CS ErrorsInSimulation

Lecture 7: Error in Simulations

Lecture 7

Bouncing Ball

Floor level

Bouncing Ball with Gravity - 1%% initializations

figure; axis equal; axis([0 100 0 100]);

curLoc = [50, 80];

curVel = [0.9, -0.1];

radius = 2;

hBall = rectangle('position', [curLoc, 2*radius, 2*radius], ...

'curvature', [1 1], 'facecolor', [0.5 1 0.9]);

%% main loop

while 1

%update position and draw

curVel(2) = curVel(2) - 0.1;

curLoc = curLoc + curVel;

set(hBall, 'position', [curLoc, 2*radius, 2*radius]);

%handle collisions

if (curLoc(1) > (100 - 2 * radius)) | (curLoc(1) < 0)

curVel(1) = -curVel(1);

end;

if (curLoc(2) > (100 - 2* radius)) | (curLoc(2) < 0)


end;

pause(0.0001);

end;

Why Does the Ball Loose Energy?

The code is written

such that the velocity of

each location is

assumed as constant for

this interval, where as in

reality it is not.

Results in incorrect

estimation



curLoc = [50, 80];

curVel = [0.9, -0.1];

radius = 2;



%% main loop

while 1





%handle collisions



end;



end;

pause(0.0001);

end;

Velocity update

Position update

Example: Update vy before y Let

y = 9 and vy = -3.9

Blow up the picture

After every 0.2 sec vy = vy -0.02 = -3.92 y = 9.00 3.92*0.2 = 8.22

vy = vy -0.02 = -3.94 y = 8.22 3.94*0.2 = 7.42

vy = vy -0.02 = -3.96 y = 7.42 3.96*0.2 = 6.34

vy = vy -0.02 = -3.98 y = 6.34 3.98*0.2 = 5.84

vy = vy -0.02 = -4.00 y = 5.84 4.00*0.2 = 5.04

Total deduction: 3.96

After 1 sec vy = vy -0.1 = -4

y = y + vy = 9 4 = 5.0

Bouncing Ball with Gravity - 1



curLoc = [50, 80];

curVel = [0.9, -0.1];

radius = 2;



%% main loop

while 1





%handle collisions



end;



end;

pause(0.0001);

end;

Position update

Velocity update

Example: Update y before vy Let

y = 9 and vy = -3.9

Blow up the picture

After every 0.2 sec vy = vy -0.02 = -3.92 y = 9.00 3.92*0.2 = 8.22

vy = vy -0.02 = -3.94 y = 8.22 3.94*0.2 = 7.42

vy = vy -0.02 = -3.96 y = 7.42 3.96*0.2 = 6.34

vy = vy -0.02 = -3.98 y = 6.34 3.98*0.2 = 5.84

vy = vy -0.02 = -4.00 y = 5.84 4.00*0.2 = 5.04

Total deduction: 3.96

After 1 sec y = y + vy = 9 3.9 = 5.1

vy = vy -0.1 = -4

Bouncing Ball with Gravity - 2

Bouncing Ball with Gravity

Update vy before y Update y before vy

vy = vy -0.1 = -4

y = y + vy = 9 4 = 5.0

y = y + vy = 9 3.9 = 5.1

vy = vy -0.1 = -4

Bouncing Ball with Gravity Let

y = 9 and vy = -3.9

After 1 sec Update vy before y

vy = vy -0.1 = -4

y = y + vy = 9 4 = 5.0

Update y before vy y = y + vy = 9 3.9 = 5.1

vy = vy -0.1 = -4

Actual y should have been y = 5.04

Mean of velocities i.e. mean of initial and final velocities vi = vy vf = vy 0.1

Vmean = (vi + vf )/2

= (vy + vy 0.1)/2

= (2vy 0.1)/2

= -3.95

y = y + vy = 7 3.95 = 5.05



curLoc = [50, 80];

curVel = [0.9, -0.1];

radius = 2;



%% main loop

while 1


curLoc = curLoc + (2*curVel - [0,0.1])/2;



%handle collisions



end;



end;

pause(0.0001);

end;

Position update

with mean velocity

Velocity update

Next: Errors in Physical Simulations

Still the simulation is not perfectly accurate

Note what happens at the time of the bounce from the floor

Floor level

Bouncing Ball

When center of ball goes outside the board (light blue), we have to test bounce from top, left,

right or bottom

Careful Bounces Left wall

At the end of a time step the ball bounces off of the left wall.

Its horizontal velocity becomes opposite of what it was

Its position also is reflected off the left wall.

Careful Bounces

Examples So new value of x should

be 28

Additional distance

wall x = 20 12 = 8

New value

wall +8 = 28

So

x = wall + (wall-x)

x = 20

x =12 x =28

Careful Bounces

Examples So new value of x should

be 8

Additional distance

wall x = 0 (-8) = 8

New value

wall + (wall x) = 0 +8 = 8

So

x = wall + (wall-x)

x = 0

x =-8 x =8

Careful Bounces Right wall

Examples Additional distance

x+2*r - wall

Updated value

wall - (x+2*r wall)

Left corner Additional -2*r

So

x = wall - (x+2*r - wall ) 2*r

Careful Bounces Left Wall

x =wall + (wall x)

Bottom Wall y =wall + (wall y)

Right wall x = wall - (x+2*r - wall ) 2*r

Top Wall y = wall (y+2*r - wall ) 2*r

Bouncing Ball Main Loopwhile 1



%handle collisions

if (curLoc(1) < 0)

curLoc(1) = -curLoc(1);


elseif (curLoc(1) > (100 - 2 * radius))

curLoc(1) = 100 - (curLoc(1) + 2*radius - 100) - 2*radius;


end;

if (curLoc(2) < 0)

curLoc(2) = - curLoc(2);


elseif (curLoc(2) > (100 - 2* radius))

curLoc(2) = 100 - (curLoc(2) + 2*radius - 100) - 2*radius;


end;


pause(0.0001);

end;

Typical Structure

Initialize graphics Display scene at t=0

Begin loop e.g. for t = 0.05:0.05:10 Compute scene at time t

pause(0.05)

Display the scene

EndTime limit of up to 1/20th of a

second for real-time graphics

For complex scenes, this could take

a large amount of time

and could vary

Better Structure

Initialize graphics

Start Stopwatch

for t = 0:0.05:10 Compute scene at time t

pause(0.05) Wait for real-time to catch up to simulation time

Display the scene

End

Better Structure

Initialize graphics

tic

for t = 0:0.05:10 Compute scene at time t

pause(t toc)

Display the scene

End

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CS ErrorsInSimulation