\documentclass{article}
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\usepackage{amsmath}
\usepackage{graphicx}

% If there are something you are wondering about, the LaTeX forum at overleaf is also very helpful. Consider LaTeX editing as if you're programming and you're using stack-overflow. Usually you will find the solution you're looking for. %

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% If you know how to type the greek symbols are, then you can simply just call them using \theta \psi and etc. Just be careful to use math-mode. Example: $\theta$ $\psi$ or $\theta \psi$

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\title{Mandatory Assignment I}
\date{January 2021}
\author{My name is ipsum Dorellium}

\begin{document}
\maketitle

\section*{Task I - Transformations}
Find $T^{B}_{T}$ (the transformation matrix expressing the position and orientation of {T} with respect to {B}, said in other words; a vector that goes from ${B}$ and ends up in ${T}$). \\

* Insert your answer here * 
\begin{equation} % Example of setting up a matrix. Use this as an template for future tasks such as in Task 2c and task 3
T^{B}_{T} = 
    \begin{bmatrix}
        R & t \\
        0 & 1 \\
    \end{bmatrix}
    =
	\begin{bmatrix}
		r_{11} &  r_{12} & r_{13} & t_{1}\\
		r_{21} &  r_{22} & r_{23} & t_{2}\\
		r_{31} &  r_{32} & r_{33} & t_{3}\\
		0 &  0 & 0 & 1\\
	\end{bmatrix}
\end{equation}

% You can add as many equations you like. Try to space them equally using separate \begin{equations} to get it to look structured and organized.
You can also write an appropriate text here to validate your answer here and then begin another environment

\begin{equation}
T^{i-1}_{i} = 
    \begin{bmatrix}
        R & t \\
        0 & 1 \\
    \end{bmatrix}
    =
	\begin{bmatrix}
		r_{11} &  r_{12} & r_{13} & t_{1}\\
		r_{21} &  r_{22} & r_{23} & t_{2}\\
		r_{31} &  r_{32} & r_{33} & t_{3}\\
		0 &  0 & 0 & 1\\
	\end{bmatrix}
	=
	\begin{bmatrix}
		r_{11} &  r_{12} & r_{13} & t_{1}\\
		r_{21} &  r_{22} & r_{23} & t_{2}\\
		r_{31} &  r_{32} & r_{33} & t_{3}\\
		0 &  0 & 0 & 1\\
	\end{bmatrix}
\end{equation}

\section*{Task II - Forward-kinematics I}
\subsection*{a)}
Sketch the workspace of the robot (a quick 3D drawing from top and side view is sufficient). \\

*Insert your image here* % Recommend using \includegraphics

\subsection*{b)}
Draw a simple 3D illustration of the robot, showing the coordinate frames and the Denavit-Hartenberg parameters. \\

*Insert your image and denavit-hartenberg parameters here* % Recommend using \includegraphics

\subsection*{c)}
Calculate the forward kinematics for this robot. Your answer should be an algebraic transformation matrix $T^{B}_{t}$ denoting the transformation of the tool coordinate frame ${t}$ located at the tip of the arm, with respect to the base coordinate frame ${B}$. This transformation matrix is a function of the three joints. \\

*Calculate the forward-kinematics using DH-convention* % Recommend using %begin{equation}


\section*{Task III - Forward-kinematics II}
Find $p^{T}$, the coordinates of point $p$ given in task coordinate frame {T}. \\

* Insert your answer here * % Recommend using %begin{equation}

\section*{Task IV - Inverse Kinematics}
\subsection*{a)}
What are the two most common ways of deriving inverse kinematics? Describe shortly how you do them, and which one you will be using in b). \\

* Insert your answer here *

\subsection*{b)}
Derive the inverse kinematic equations for the simplified robot, and show the different steps. \\ % Recommend using %begin{equation}

* Insert your calculations here *
% You don't have to use equation environment for writing your answers. You can also use just $$ to activate math-mode. Look below. 
% However, using environments usually tends to structure the equations in a proper format, so I highly recommend using environment where it is possible. I have commented both of the methods. Choose the one that suits to your style. 

% Method I - These methods are NOT part of the solutions of the ACTUAL task. This method serves as an example. 
\begin{equation} 
    \begin{aligned}
        \theta_1 = x+y+z
    \end{aligned}
\end{equation}

\begin{equation} 
    \begin{aligned}
        \theta_{2} = \dfrac{\sqrt{12}}{\tan{50}}
    \end{aligned}
\end{equation}

% Method II - These methods are NOT part of the solutions of the ACTUAL task. This method serves as an example. 
$\theta_{1} = x+y+z $ \\ 
$\theta_{2} = \dfrac{\sqrt{12}}{\tan{50}}$ \\

\subsection*{c)}
How many solutions exist for the joint angles given an arbitrary position of the tip of link 3? We do not have any physical restriction on θ1, and
you can also assume that there is no physical restriction on θ2 other than the angles that would place $L_2$ right on top of $L_1$. \\

* Insert your answer here *  


\end{document}
