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\usepackage{tkz-euclide} -\usetikzlibrary{external,shapes,through} +\usetikzlibrary{external,shapes,through,arrows} \tikzexternalize[prefix=figures/] \tikzstyle{none}=[] \input{./tikzs/styles.tikzstyles} +\tikzset{>=stealth} + \pgfdeclarelayer{nodelayer} \pgfdeclarelayer{edgelayer} \pgfsetlayers{edgelayer,nodelayer,main} \graphicspath{{../figures/}{./figures/}} - \newcommand*{\vv}[1]{\overrightarrow{#1}} - - \newcommand*{\figref}[1]{\figurename~\ref{#1}} \captionsetup{labelfont={bf}} diff --git a/sections/appendix.tex b/sections/appendix.tex index ef4ab9f..bd039b9 100644 --- a/sections/appendix.tex +++ b/sections/appendix.tex @@ -3,7 +3,7 @@ \section{Calculations} \label{appendix:calculations} -\paragraph{Moment of inertia of rectangle} +\subsection{Moment of inertia of rectangle} \begin{equation} \label{eq:rect_moment_long} \begin{split} @@ -22,7 +22,7 @@ \end{equation} \newpage -\paragraph{Moment of inertia of sub-triangle of regular polygon} +\subsection{Moment of inertia of sub-triangle of regular polygon} Before starting the calculations, it is to be noted that according to Figure \ref{fig:subtriangle}, we have that $$ \tan\left(\frac{\theta}{2}\right) = \frac{\frac{l}{2}}{h} = \frac{l}{2h} $$ @@ -46,7 +46,7 @@ it will be useful to simplify the result of the integral. \end{equation} \newpage -\paragraph{Moment of inertia of sub-triangle of arbitrary polygon} Recall +\subsection{Moment of inertia of sub-triangle of arbitrary polygon} Recall equation \ref{eq:r} defines $$ \vec r = \alpha \vv{CA} + \beta \alpha \vv{AB} $$ \begin{equation} @@ -62,3 +62,10 @@ $$ \vec r = \alpha \vv{CA} + \beta \alpha \vv{AB} $$ &= \frac{\rho hb}{4} \left(\frac{1}{3}\vv{AB}^2 + \vv{AB} \cdot \vv{CA} + \vv{CA}^2\right) \\ \end{split} \end{equation} + +\newpage +\subsection{Solving for impulse parameter} +\label{app:impulse_long} +[to be done :)] +\begin{equation} +\end{equation} diff --git a/sections/theoretical_background.tex b/sections/theoretical_background.tex index 6180fe2..c612586 100644 --- a/sections/theoretical_background.tex +++ b/sections/theoretical_background.tex @@ -6,6 +6,7 @@ polygons; the resolution of the collision, i.e. finding the final velocity vectors and angular speed of those polygons. \subsection{Moment of inertia} +\label{sub:moment} The inertia of an object refers to the tendency of an object to resist a change of its state of motion or rest, it describes how the object behaves when forces @@ -264,6 +265,7 @@ to the overall polygon. where, $P_{n+1} = P_1$ in the case of $i = n$. \subsection{Collision detection} +\label{sub:collision-detection} Collision detection, as the name suggests, are the algorithms used to detect whether two polygons are colliding. The result of this procedure must be an @@ -462,3 +464,161 @@ find the normal. The results look realistic enough to be accepted. \subsection{Collision resolution} \label{sub:resolution} + +The collision resolution is the last step in the processing of the collision. +Algorithmically, it is much less heavy than collision detection, since, once the +simulation has two colliding polygons, a point of impact and a normal vector, +it's just a case of applying the rigid body physics formulas to the polygons +that are colliding. This part has been helped a lot by the works of Erik Neumann +\cite{collision:resolution-site} and Chris Hecker +\cite{collision:resolution-paper}. + +\begin{figure}[H] + \centering + \inputtikz[.5]{collision_resolution} + \caption{Collision resolution between polygons $A$ and $B$} + \label{fig:collision_resolution} +\end{figure} + +\paragraph{Variable definition} Before getting into any maths, let's define some +variables that we are going to use + +\begin{multicols}{2} + \begin{itemize} + \item $m_a, m_b = $ mass of the bodies $A$ and $B$ + \item $\vec r_{ap} = $ distance vector from center of mass of body $A$ to + point $P$ + \item $\vec r_{bp} = $ distance vector from center of mass of body $A$ to + point $P$ + \item $\omega_{a1}, \omega_{b1} = $ initial angular velocity of bodies + $A, B$ + \item $\omega_{a2}, \omega_{b2} = $ final angular velocity of bodies + $A, B$ + \item $\vec v_{a1}, \vec v_{b1} =$ initial velocities of center of mass + bodies $A, B$ + \item $\vec v_{a2}, \vec v_{b2} =$ final velocities of center of mass + bodies $A, B$ + \item $\vec v_{ap1}=$ initial velocity of impact point $P$ on body $A$ + \item $\vec v_{bp1}=$ initial velocity of impact point $P$ on body $B$ + \item $\vec v_{p1}=$ initial relative velocity of impact points on body $A, B$ + \item $\vec v_{p2}=$ final relative velocity of impact points on body $A, B$ + \item $\vec n=$ normal vector + \item $ e=$ elastic coefficient (0 = inelastic, 1 = perfectly elastic) + \end{itemize} +\end{multicols} + +Some of those variables, like the ones in the left column, are already given by +the simulation, some of them have been computed thanks to the algorithms in +section \ref{sub:collision-detection} (normal vector and position of point $P$), +and some have still to be defined mathematically, such as $\vec v_{ap1},\vec + v_{bp1},\vec v_{p1}$ and $\vec v_{p2}$. So the velocity vectors of impact +point $P$ on both bodies, before the collision, are +\begin{equation} \label{eq:vabp1} + \begin{split} + \vec v_{ap1} = \vec v_{a1} + \omega_{a1} \times \vec r_{ap} \\ + \vec v_{bp1} = \vec v_{b1} + \omega_{b1} \times \vec r_{bp} + \end{split} +\end{equation} + +Similarly, the final velocities are +\begin{equation} \label{eq:vabp2} + \begin{split} + \vec v_{ap2} = \vec v_{a2} + \omega_{a2} \times \vec r_{ap}\\ + \vec v_{bp2} = \vec v_{b2} + \omega_{b2} \times \vec r_{bp} + \end{split} +\end{equation} +Here we are regarding the angular velocity as a 3-dimensional vector +perpendicular to the plane, so that the cross product is calculated as + +$$ \omega \times \vec r = \begin{pmatrix} 0\\0\\\omega \end{pmatrix} \times + \begin{pmatrix} r_x\\r_y\\0 \end{pmatrix} = \begin{pmatrix} -\omega r_y \\ + \omega r_x \\0\end{pmatrix} $$ + +We these variables, we can finally define the relative velocities $\vec + v_{p1}$ and $\vec v_{p2}$ + +\[ \begin{split} + \vec v_{p1} = \vec v_{ap1} - \vec v_{bp2}\\ + \vec v_{p2} = \vec v_{ap2} - \vec v_{bp2} + \end{split} \] +If we expand by using \ref{eq:vabp1} and \ref{eq:vabp2}, we get +\begin{equation} + \begin{split} + \vec v_{p1} = \vec v_{a1} + \omega_{a1} \times \vec r_{ap} - \vec v_{b1} + + \omega_{b1} \times \vec r_{bp}\\ + \vec v_{p2} = \vec v_{a2} + \omega_{a2} \times \vec r_{ap} - \vec v_{b2} + \omega_{b2} \times \vec r_{bp} + \end{split} +\end{equation} + +The relative velocity of point $P$ along the normal vector $\vec n$ is +$$ \vec v_{p1} \cdot \vec n $$ +Note that for a collision to occur this relative normal velocity must be +negative (that is, the objects must be approaching each other). Let e be the +elasticity of the collision, having a value between 0 (inelastic) and 1 +(perfectly elastic). We now make an important assumption in the form of the +following relation +\begin{equation} + \label{eq:vp2n} + \vec v_{p2} \cdot \vec n = - e \vec v_{p1} \cdot \vec n +\end{equation} +This says that the velocity at which the objects fly apart is proportional to +the velocity with which they were coming together. The proportionality factor is +the elasticity $e$. + +\paragraph{Collision Impulse} In simple terms, collision impulse refers to the +change in momentum experienced by an object during a collision. It is a measure +of the force applied to an object over a short period of time. We imagine that +during the collision there is a very large force acting for a very brief period +of time. If you integrate (sum) that force over that brief time, you get the +impulse. + +In our simulation, we are assuming that no friction is happening during the +collision, so that the only force we have to consider is the one of along the +normal vector $\vec n$. The friction would create a force perpendicular to the +normal, and it would make things a little too complicated for the scope of this +project. + +Since the only force we consider is the normal one, we can consider the net +impulse to be $j \vec n$, where $j$ is the impulse parameter. The body $A$ will +experience the net impulse $j \vec n$ and body $B$ will experience it's negation +$- j \vec n$ since the force that $B$ experience is equal an opposite to the one +experienced by $A$. The impulse is a change in momentum. Momentum has units of +velocity times mass, so if we divide the impulse by the mass we get the change +in velocity. We can relate initial and final velocities as + +\begin{equation} + \label{eq:va2} + \vec v_{a2} = \vec v_{a1} + \frac{j\vec n}{m_a} +\end{equation} +\begin{equation} + \label{eq:vb2} + \vec v_{b2} = \vec v_{b1} - \frac{j\vec n}{m_b} +\end{equation} + + +For the final angular speed, it's change from the impulse $j\vec n$ is given by +$\vec r_{ap} \times j \vec n$. We can divide the result by the moment of +inertia, which was calculated in section \ref{sub:moment}, to get convert the +change in angular momentum into change in angular speed. Similarly as above, we +can relate the initial and final angular velocity as + +\begin{equation} + \label{eq:omega_a2} + \omega_{a2} = \omega_{a1} + \frac{\vec r_{ap} \times j\vec n}{I_a} +\end{equation} +\begin{equation} + \label{eq:omega_b2} + \omega_{b2} = \omega_{b1} - \frac{\vec r_{bp} \times j\vec n}{I_b} +\end{equation} + +\paragraph{Solving for the impulse parameter} +Now we have everything we need to solve for the impulse parameter $j$. The bulk +of the calculations can be found in section \ref{app:impulse_long} of the +appendix \ref{appendix:calculations}. But basically we start with equation +\ref{eq:vp2n} and we end up with +\begin{equation} + \label{eq:j} + j = \frac{ - (1+e) \cdot \vec v_{ap1} \cdot \vec n }{\frac{1}{m_a} + \frac{1}{m_b} + + \frac{\left( \vec r_{ap} \times \vec n \right)^2}{I_a} + \frac{\left( \vec + r_{bp} \times \vec n \right)^2}{I_b}} +\end{equation}