A Brief Introduction to Animation with MATLAB

Dmitry Savransky (dsavrans@princeton.edu) August 9, 2010

Basic Animation Concepts

In this section, we will cover the basics needed to create simple MATLAB animations analogous to the traditional cel animation techniques used in analog animation. This process essentially involves displaying slightly varying images quickly enough so that the brain is able to stitch them together into a seamless illusion of motion. To do this, we need to know: Dynamics 2D graphics Flow control

1.1

Dynamics

The reason we include dynamics in the list of topics required for animation, is that we need a way to describe how the objects were animating are moving. We do this by guring out the physics of our system, and solving for the equations of motion - topics that land squarely within the eld of dynamics. Before you can animate a system, you have to identify its degrees of freedom, dene coordinates to describe these, and nd the trajectories of these coordinates for some initial conditions. Usually, this will involve numerical integration, but, occasionally, you can nd analytical solutions for the trajectories of simpler systems. As a working example, lets consider the springpendulum, as shown in Figure 1. This system is composed of a point mass m and a spring with spring constant k and rest length r0 which is constrained to swing within a plane. The equations of motion are found quite simply from Newtons second law and are given by g sin = 2r r r 2 r = g cos + r k (r r0 ) m (1.1) (1.2)

which we transform into the rst order system of ODEs by dening the state X r r
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An Introduction to MATLAB

Figure 1: Reference Frames and FBD for the spring-pendulum. whose rst derivative is given by r k 2 r + g cos (r r0 ) m = X 2r g sin r r

X2

. (1.4)

k 2 X2 X4 + g cos X3 (X1 r0 ) m = X4 2X2 X4 g sin X3 X1 X1

Having found the equations of motion, we now need to numerically integrate them to nd the system trajectory. We use MATLABs ODE45 since it is able to converge for most systems. Of course, sometimes, youll want to use one of the other MATLAB integrators, or even one of your own, but ode45 is usually a good place to start. We will also use the nested function method for passing parameters to our integrator function (like k, m, and r0 ). We simply place our integrator function inside of another function in which these variables are dened, as in Listing 1. Listing 1: Spring-Pendulum integrator.
function [t , X ] = spring_pendulum_int ( tspan , x0 , l_0 ,m , k ) %set constants and defaults i f ~ e x i s t ( l_0 , var ) , l_0 = 0.5; end i f ~ e x i s t ( m , var ) , m = 1; end i f ~ e x i s t ( k , var ) , k = 5; end g = 9.81; %acceleration due to gravity m/s^2 %set initial conditions and time span i f ~ e x i s t ( x0 , var ) , x0 = [1 ,0 , pi /4 ,0]; end i f ~ e x i s t ( tspan , var ) , tspan = [0 ,20]; end %integrate equations of motion [t , X ] = ode45 ( @springpendulum_eq , tspan , x0 );

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%integrator for equations of motion function dx = springpendulum_eq (t , x ) %y = [r, rd, th, thd] r = x (1); rd = x (2); th = x (3); thd = x (4); dx = [ rd ; r * thd ^2 + g * cos ( th ) - k / m *( r - l_0 ); thd ; -2* rd * thd / r - g * s i n ( th )/ r ]; end
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end

Note that we now have a function which will take a time array, initial conditions, and the three spring-pendulum parameters, and return the spring-pendulum trajectory.

1.2

2D Graphics

The two basic commands we use for 2D graphics are plot and fill. The former creates 2D scatter and line plots, the latter creates 2D polygons. Both take as inputs arrays of x and y Cartesian coordinates (fill also requires an input specifying color). Example 1.1. Plot shaded and unshaded squares and circles First, we dene the coordinates for a unit circle and square: theta = xcirc = ycirc = xsquare ysquare 0:pi/100:2*pi; sin(theta); cos(theta); = [-1 -1 1 1 -1]; = [-1 1 1 -1 -1];

Note that we dened the circle parametrically (via the polar coordinate ). Also notice that we need to dene 5 points for the square to produce a closed gure. Now, lets draw some shapes: fill(5*xsquare,5*ysquare,r); axis square hold on fill(4*xcirc, 4*ycirc,b); plot(2*xsquare,2*ysquare,g--); plot(xcirc,ycirc,c,LineWidth,2); hold off

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Example 1.1 demonstrates how to draw basic geometric shapes. There are four very important things to note here: 1. To draw a geometric shape, you only need to identify its vertices. The appearance of smooth curves (like circles) will depend on how nely you parametrize the shape. 2. The default axes MATLAB creates when you plot something will not have an aspect ratio that makes pixels the same length in the x and y dimension. To x this, you need to use the axis command. 3. To put multiple dierent objects on the same plot, we use the hold command. 4. Both plot and fill can be modied with numerous optional property settings, like LineWidth. For a complete list of these, look in the MATLAB documentation. Almost any dynamical system can be represented as a collection of lines and basic shapes. For our spring-pendulum example, we can use as little as two: a line representing the spring (pendulum arm) and a circle for the pendulum bob. We can also add an element (for example, a square) to represent the pendulum attachment point. Example 1.2. Draw a pendulum Well use elements from example 1.1 to draw a pendulum with r = 10 and = 45 . theta = 45*pi/180; r = 10; xp = r*sin(theta); yp = -r*cos(theta); fill(xsquare,ysquare,g); hold on fill(xcirc+xp, ycirc+yp,b); plot([0,xp],[0,yp],r); axis square To properly orient everything in your gures, it is important to remember that MATLAB denes the positive x and y directions to the right and up, respectively. When creating gures, its very important to take into account how the reference frame we were working with when setting up our dynamics problem corresponds to MATLABs default axes. For the spring-pendulum, MATLABs gure axes correspond to (e1 , e2 ) as shown in Figure 1, whereas we dened our pendulum position in the frame (er , e ) as rer . To properly draw our results, we need a transformation table for the two frames er e e1 sin cos e2 cos sin (1.5)

This table gives us the equations used to transform r and into Cartesian coordinates used in example 1.2.

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1.3

Flow Control

Flow control refers to the ability to repeat commands multiple times or to exhibit dierent behaviors depending on the values of some variables. Repeating commands is known as looping, and MATLAB contains two types of loops: for and while A for loop executes a group of commands a specied number of times and is written as: for k = someVector % your code here, possibly depending on k end % more code k will take on the value of each member of someVector (which can be any vector, or operation resulting in a vector), and run through all the code up to end using that value. This process will continue until it runs out of elements in someVector, at which point it will exit the loop and move on to the rest of the code. Thus, a for loop runs as many times as there are elements in someVector. We can use a forloop to draw our spring-pendulum system at each time step of the integrator, thus producing motion, but we need one more element: the pause command. This tells MATLAB to wait for a given amount of time, allowing us to actually perceive the motion (otherwise, all of the images would ash by too fast to see). Putting all of this together results in the code in Listing 2. Listing 2: Spring-Pendulum animator.
function spring_pendulum_anim ( varargin ) %integrate equations of motion [t , X ] = spring_pendulum_int ( varargin {:});
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%extract length and angle in time r = X (: ,1); th = X (: ,3);

%find position of the pendulum bob: rm = [ r .* s i n ( th ) , -r .* cos ( th )]; %find bounds: maxr = max( abs ( r )); xmin = min ( rm (: ,1)) - maxr /10; xmax = max( rm (: ,1))+ maxr /10; ymin = min ( rm (: ,2)) - maxr /10; ymax = max([0 ,max( rm (: ,2))])+ maxr /10; %create and clear figure h = f i g u r e (1); c l f ( h ); % create a circle and a square: a = 0: pi /100:2* pi ;

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s = [ -1 -1 1 1 -1; -1 1 1 -1 -1]; xscirc = yscirc = xssquare yssquare maxr /50* s i n ( a ); maxr /50* cos ( a ); = maxr /50* s (1 ,:); = maxr /50* s (2 ,:);

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%step in time: f o r i =1: length ( t ) %plot track so far: plot ( rm (max([ i -100 ,1]): i ,1) , rm (max([ i -100 ,1]): i ,2) , k - - ); hold on %connect mass and pivot: plot ([0 , rm (i ,1)] ,[0 , rm (i ,2)] , r , LineWidth ,2) %draw the pivot (centered at origin) and the mass: f i l l ( xssquare , yssquare , g ); f i l l ( xscirc + rm (i ,1) , yscirc + rm (i ,2) , b );

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%set axes to proper values: a x i s equal ; grid on ; a x i s ([ xmin xmax ymin ymax ]); hold off ; %pause for length of time step i f i < length ( t ) pause ( t ( i +1) - t ( i )); end end

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Notice how we used the actual distance between integrator time steps in our pausecommandto give our animation a realistic progression of time.

Animation via Graphics Objects

Every time you create something in a MATLAB gure, whether it is a line, or lled polygon, or even a new set of axes, you are actually creating an object with specic attributes. All MATLAB objects can be referred to via their handles, which gives us the ability to both get and set an objects properties. This allows for a dierent type of animation - one where we create the set of objects needed to describe our system, and then update the positions/appearences of those that represent moving parts of the system. This can be much more ecient than the method described in section 1, since it often requires less redrawing. Every command weve used so far can actually be called with an output, which will return the handle of the created object. We can then access these objects properties via the get command, and change them with the set command.

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Example 2.1. Creating and handling objects Well use elements from example 1.1 to create and manipulate a ll object: clf h1 = fill(xsquare,ysquare,g); axis([-5 5 -5 5]); get(h1) The last command will produce a list of all of the properties of our ll object. Of particular interest are the XData and YData properties, which control the objects location within the axes. We can change these properties to move the object: set(h1,XData,xsquare+3,YData,ysquare+3); Notice that we didnt need to reset our axes, since we didnt add any new graphics objects, but merely moved an existing one. This is another way in which this method is more ecient. We can modify our rst animation to use this technique, resulting in the code in Listing ??. Listing 3: Better Spring-Pendulum animator.
function spr ing_ pend ulum_ anim 2 ( varargin ) %integrate equations of motion [t , X ] = spring_pendulum_int ( varargin {:});
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%extract length and angle in time r = X (: ,1); th = X (: ,3);

%find position of the pendulum bob: rm = [ r .* s i n ( th ) , -r .* cos ( th )]; %find bounds: maxr = max( abs ( r )); xmin = min ( rm (: ,1)) - maxr /10; xmax = max( rm (: ,1))+ maxr /10; ymin = min ( rm (: ,2)) - maxr /10; ymax = max([0 ,max( rm (: ,2))])+ maxr /10; %create and clear figure h = f i g u r e (1); c l f ( h ); % create a circle and a square: a = 0: pi /100:2* pi ; s = [ -1 -1 1 1 -1; -1 1 1 -1 -1]; xscirc = yscirc = xssquare yssquare maxr /50* s i n ( a ); maxr /50* cos ( a ); = maxr /50* s (1 ,:); = maxr /50* s (2 ,:);

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%create the pendulum components %draw the pivot (centered at origin) and the mass: f i l l ( xssquare , yssquare , g ); hold on bob = f i l l ( xscirc + rm (1 ,1) , yscirc + rm (1 ,2) , b ); %connect mass and pivot: arm = plot ([0 , rm (1 ,1)] ,[0 , rm (1 ,2)] , r , LineWidth ,2); %plot track so far: track = plot ( rm (1 ,1) , rm (1 ,2) , k - - );

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%set axes to proper values: a x i s equal ; grid on ; a x i s ([ xmin xmax ymin ymax ]); %step in time: f o r i =1: length ( t ) %update all moving elements s e t ( bob , XData , xscirc + rm (i ,1) , YData , yscirc + rm (i ,2)); s e t ( arm , XData ,[0 , rm (i ,1)] , YData ,[0 , rm (i ,2)]); s e t ( track , XData , rm (max([ i -100 ,1]): i ,1) , ... YData , rm (max([ i -100 ,1]): i ,2)); %pause for length of time step i f i < length ( t ) pause ( t ( i +1) - t ( i )); end end

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