Update octave dan berhasil create plot ulang seperti di jurnal. Menambahkan documentasi untuk mengingat secara singkat
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% ASSIGNMENT INFORMATION
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%----------------------------------------------------------------------------------------
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\title{Controlling a Triangular Formation of Mobile Agent} % Title of the assignment
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\title{Kendali Proposional-Integral Pada Pelacakan Formasi Berdasarkan Jarak} % Title of the assignment
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\author{Anggoro Dwi Nur Rohman\\ \texttt{anggoro\_dwi@student.ub.ac.id}} % Author name and email address
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@ -38,42 +38,120 @@
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%----------------------------------------------------------------------------------------
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\begin{document}
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\maketitle % Print the title
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\section*{Pendahuluan}
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Akan dirangkum dari penelitian
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Rangkuman ini ditujukan untuk memahami dan review jurnal. Judul dalam bahasa inggris
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\textit{A Proportional-Integral Controller for Distance-Based Formation Tacking} yang
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diteliti oleh Oshri Rozenheck dari Israel Institute of Technology, Haifa, Israel.
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Peneliti menerangkan pada jurnal ini tentang permasalahan kendali formasi pada
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multi-agent apabila pada salah satu agent nya diberikan kecepatan tambahan sebagai
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refrensi.
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\section{Formasi Segitiga}
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%% \begin{table}
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\section{Notasi-Notasi}
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%% \caption{Notasi-notasi}
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\begin{tabular}{| l | l |}
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\hline
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Notasi & Keterangan \\ \hline
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$n \triangleq |V|$ & ... \\ \hline
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Robot ditandai dengan 1,2,3. Apabila robot 1 mengikuti robot 2 maka dinotasikan dengan $[1] = 2$. jarak antara $i$ dan $[i]$ dinotasikan $d_i$.
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$m \triangleq |\varepsilon| $ & .. \\ \hline
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Koordinat vector dari agent $i$ dinotasikan dengan $x_i$ terhadap global koordinat yang fiks, dan $y_{ij}$ adalah posisi robot $j$ terhadap sistem koordinat dari $i$ yang telah tentukan. Apabila $R_i$ dan $\tau_i$ adalah matriks rotasi dan vector translasi maka $y_{ij} = R_ix_j + \tau_i, j \in \{1,2,3\}$. Penelitian ini menggunakan kinematik yang sedarhana
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\begin{eqnarray*}
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\dot{y}_{ii} &=& u_i \quad i \in \{1,2,3\} \\
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\dot{x}_{i} &=& R_i^{-1} u_i
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\end{eqnarray*}
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$A_1, \dots, A_n \in \mathbb{R}^{p\times q}$ & ... \\ \hline
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\section*{Referensi}
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%% \printbibliography
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$x = \begin{bmatrix} x_1^T \\ \dots\\ x_n^T \end{bmatrix} \in \mathbb{R}^{2n}$ &
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Konfigurasi titik \\
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$x_i \in \mathbb{R}^n$ & \\
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$ x_i \neq x_j ; \forall i \neq j$ & \\ \hline
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%% \begin{refsection}
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%% @INPROCEEDINGS{Cao2007,
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%% author={M. {Cao} and A. S. {Morse} and C. {Yu} and B. D. O. {Anderson} and S. {Dasguvta}},
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%% booktitle={2007 46th IEEE Conference on Decision and Control},
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%% title={Controlling a triangular formation of mobile autonomous agents},
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%% year={2007},
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%% volume={},
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%% number={},
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%% pages={3603-3608},
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%% abstract={This paper proposes a distributed control law for maintaining a triangular formation in the plane consisting of three mobile autonomous agents. It is shown that the control law can cause any initially non-collinear, positively-oriented {resp. negatively-oriented} triangular formation to converge exponentially fast to a desired positively-oriented {resp. negatively- oriented} triangular formation. It is also shown that there is a thin set of initially collinear formations which remain collinear and may drift off to infinity as t rarr infin. These findings complement and extend earlier findings cited below.},
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%% keywords={distributed control;mobile robots;multi-robot systems;spatial variables control;triangular formation;mobile autonomous agents;collinear formations;distributed control law;Autonomous agents;USA Councils;Distributed control;H infinity control;Differential equations;Information technology;Art;Australia Council},
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%% doi={10.1109/CDC.2007.4434757},
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%% ISSN={0191-2216},
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%% month={Dec},}
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%% \end{refsection}
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$e_k \triangleq x_j - x_i$ & Relative position vector \\ \hline
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$e = \begin{bmatrix} e_1^T \\ \vdots \\ e_m^T \end{bmatrix} \in \mathbb{R}^{2m}$ & Edge Vector \\
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$E \in R^{n\times m}$ & Incidence matrix dimana isinya adalah $\{0, \pm 1\}$ \\
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& barisnya menandakan vertices dan kolom nya menandakan edge \\ \hline
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\hline
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\end{tabular}
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%% \end{table}
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\section{ Edge Function}
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Diketahui framework $G(x)$ dengan vector edge ${e}_{k=1}^m$,
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$F : \mathbb{R}^{2n} \times G \rightarrow \mathbb{R}^m$, maka dapat didefinisi
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\begin{equation}
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F(x,G) \triangleq = \begin{bmatrix}|e_1||^2 \\ \vdots \\ ||e_m||^2 \end{bmatrix}
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\end{equation}
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Peneliti mendefinisi \textit{rigidity matrix R(x)} sebagai fungsi jacobian dari
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Edge function,
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\begin{equation*}
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R(x) = \frac{\partial F(x,G)}{\partial x} \in \mathbb{R}^{m \times 2n}
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\end{equation*}
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dan menyingkat perhitungannya dan menghasilkan persamaan
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\begin{equation}
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R(x) = diag(e_i^T)(E^T \otimes I_2)
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\end{equation}
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\section{Kendali}
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Error jarak dinotasikan dengan $\delta \in \mathbb{R}^m$ dimana
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\begin{equation}
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\delta_k = || e_k ||^2 - d_k^2, k \in \{1, \dots, m\}
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\end{equation}
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Peneliti menggunakan fungsi potensial
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\begin{equation}
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\Phi(e) = \frac{1}{2} \sum_{k-1}^{m} \Big( || e_k ||^2 - d_k^2 \Big)^2 =
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\frac{1}{2} \sum_{k=1}^{m} \delta_k^2
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\end{equation}
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digunakan untuk kendali pada setiap robot dengan cara negative gradient
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dari fungsi potensial.
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\begin{equation}
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u_i(t) = - \Big(\frac{\partial \Phi(e)}{\partial x_i} \Big) =
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- \sum_{j ~ o} \Big( || e_k ||^2 - d_k^2 \Big) e_k
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\end{equation}
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Apabila ditulis ulang dalam bentuk state-space
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\begin{equation}
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\dot{x}(t) = - R(x)^T R(x) x(t) + R(x)^Td
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\end{equation}
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\begin{figure}
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\centering
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\includegraphics[scale=.7]{./IMG/fig2.png}
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\caption{Skema General}
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\label{f1}
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\end{figure}
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Menggunakan skema seperti pada gambar.\ref{f1} maka dapat diperoleh
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\begin{eqnarray}
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\dot{x} &=& u(t) + B.v_{ref} \\
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u(t) &=& -R(x)^T.C.\Big(R(x).x(t) - d \Big) \label{controller}
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\end{eqnarray}
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dan cikal bakal $C$ sebagai controller.
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dengan menerapkan Proportional-integrator pada persamaan \eqref{controller}
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maka diperoleh
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\begin{equation}
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u(t) = -R(x)^T k_p ( R(x).x(T) - d ) -
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R(x)^T.K_i \int_0^T (R(x).x(\tau) - d) d \tau
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\end{equation}
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dimana $kp_p$ dan $k_i$ adalah konstanta.
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Pada bagian integrator menghasilkan state baru
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\begin{equation}
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\dot{\xi} = k_i(R(x).x(t) -d)
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\end{equation}
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lalu kombinasi persamaan tersebut diperoleh state- space close loop
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\begin{equation}
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\begin{bmatrix} \dot{x}(t) \\ \dot{\xi(t)} \end{bmatrix} =
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\begin{bmatrix} -k_p.R(x)^T.R(x) & -R(x)^T \\ k_i.R(x) & 0 \end{bmatrix}
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\begin{bmatrix} x(t) \\ \xi(t) \end{bmatrix} +
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\begin{bmatrix} k_p.R(x)^T \\ -k_i.I \end{bmatrix}.d +
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\begin{bmatrix} B \\ 0 \end{bmatrix}.v_{ref}
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\end{equation}
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\subsection{Implementasi}
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Pada seksi implementasi ini akan dibahas mengenai code yang digunakan
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untuk mensimulasi persamaan yang diciptakan oleh peneliti.
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\end{document}
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@ -0,0 +1,92 @@
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clear -all
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close all
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%% conRobot = [2 1 1; 3 2 1; 3 1 1;3 4 1; 2 4 1; 5 3 1; 5 1 1;6 4 1; 6 2 1; 5 6 1];
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conRobot = [1 2 1;
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1 4 1;
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1 5 1;
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1 6 1;
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2 3 1;
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2 4 1;
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3 4 1;
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4 5 1;
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5 6 1;];
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%% conRobot = [2 1 1; 3 2 1; 3 1 1;3 4 1; 2 4 1;];
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%% corRobot = [0; 0; 1; 0; 2; 2; 1; 1; -1; -1; 0; -1; ]; % xy xy xy
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corRobot = [2; 2.5;
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2.7; 2.6;
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3; 2.3;
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2.5; 1.5;
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1.8; 1.5;
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1.5; 2;
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]; % xy xy xy
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%% corRobot = [0; 0; 1; 0; 2; 2; 1; 1;]; % xy xy xy
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d = 3;
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%% figure(1)
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[R,K] = rigidityMatrixFnc(conRobot);
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vref = [-5; 5];
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B = [1 0;0 1; 0 0;0 0; 0 0;0 0; 0 0; 0 0; 0 0; 0 0; 0 0; 0 0;];
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kp = 2;
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ki = 3;
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sInit = [corRobot; zeros(size(R(corRobot,K),1),1)];
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tspan = linspace(1,30);
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dydt = @(t, y) systm_robot(t, y,d, R,K, kp, ki, B, vref);
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[t,y] = ode45(dydt, tspan, sInit);
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%% figure(2)
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str_tmp = "plot(";
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for i = 1:length(corRobot)-1
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str_tmp = strcat(str_tmp,sprintf("y'(%i,:),",i));
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endfor
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str_tmp = strcat(str_tmp,sprintf("y'(%i,:));",i+1));
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eval(str_tmp);
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hold on
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str_tmp = " plot_rb = @(t) plot( [";
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for i = 1:2:length(corRobot)
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str_tmp = strcat( str_tmp, sprintf("y'(%i,t), ",i));
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endfor
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str_tmp = strcat( str_tmp, sprintf("], ["));
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for i = 2:2:length(corRobot)
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str_tmp = strcat( str_tmp, sprintf("y'(%i,t), ",i));
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endfor
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str_tmp = strcat( str_tmp, sprintf("], \"^\");"));
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eval(str_tmp)
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xrb = 1:2:length(corRobot);
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yrb = 2:2:length(corRobot);
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t = 1;
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plot_rb(t);
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for i = 1:length(conRobot)
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plot([y'(xrb(conRobot(i, 1)),t), y'(xrb(conRobot(i, 2)),t)],
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[y'(yrb(conRobot(i,1)),t), y'(yrb(conRobot(i,2)),t)], "r" )
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endfor
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t = 20;
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plot_rb(t);
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for i = 1:length(conRobot)
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plot([y'(xrb(conRobot(i, 1)),t), y'(xrb(conRobot(i, 2)),t)],
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[y'(yrb(conRobot(i,1)),t), y'(yrb(conRobot(i,2)),t)], "r" )
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endfor
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t = 40;
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plot_rb(t);
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for i = 1:length(conRobot)
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plot([y'(xrb(conRobot(i, 1)),t), y'(xrb(conRobot(i, 2)),t)],
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[y'(yrb(conRobot(i,1)),t), y'(yrb(conRobot(i,2)),t)], "r" )
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endfor
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t = 80;
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plot_rb(t);
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for i = 1:length(conRobot)
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plot([y'(xrb(conRobot(i, 1)),t), y'(xrb(conRobot(i, 2)),t)],
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[y'(yrb(conRobot(i,1)),t), y'(yrb(conRobot(i,2)),t)], "r" )
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endfor
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%% plot_rb(20);
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%% plot_rb(40);
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%% plot_rb(60);
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%% plot_rb(80);
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%% plot_rb(100);
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@ -1,18 +0,0 @@
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clear -all
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conRobot = [2 1 1; 3 2 1; 3 1 1;3 4 1; 2 4 1];
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corRobot = [0; 0; 1; 0; 2; 2; 1; 1; ]; % xy xy xy
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d = 3;
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figure(1)
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[R,K] = rigidityMatrixFnc(conRobot);
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vref = 5;
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B = [1;1; 0;0; 0;0; 0; 0;];
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kp = 2;
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ki = 10;
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sInit = [corRobot; 0;0;0;0;0;];
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tspan = linspace(1,10);
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dydt = @(t, y) systm_robot(t, y,d, R,K, kp, ki, B, vref);
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[t,y] = ode45(dydt, tspan, sInit);
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figure(2)
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plot(y'(1,:),y'(2,:),y'(3,:),y'(4,:),y'(5,:),y'(6,:),y'(7,:),y'(8,:))
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Binary file not shown.
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@ -6,7 +6,7 @@ function [R,K] = rigidityMatrixFnc (edgeL)
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% generasi koneksi bentuk incidence
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tmp = edgeL2adj(edgeL);
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gInc = adj2inc(tmp);
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drawCircGraph(tmp)
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%% drawCircGraph(tmp)
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tmp = getNodes(gInc,'adjacency')
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% generasi vector error menggunakan matrix
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% koneksi
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@ -5,10 +5,13 @@ function dxdt = systm_robot(t,x,d,R,K,kp,ki,B,vref)
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dxdt = zeros(l,1);
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x1 = x(1:n*2);
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x2 = x(length(x1)+1:length(x));
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if t > 5
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B = B * 0;
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endif
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%% if (t > 5) && (t < 15)
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%% vref = [5; 10];
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%% else
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%% if (t > 15) && (t > 20)
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%% vref = [5; -10];
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%% endif
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%% endif
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dxdt(1:length(x1)) = -((kp*R(x1,K)'*R(x1,K))*x1)-(R(x1,K)'*x2)+((kp*R(x1,K)')*(ones(size(R(x1,K)',2),1)*d))+(B*vref);
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%% ki*R(x1,K)*x1-(ki*ones(m,1))*d;
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dxdt(length(x1)+1:length(x)) = ki*R(x1,K)*x1-(ki*ones(m,1))*d;
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endfunction
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