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BoxSimulation.html
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BoxSimulation.html
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<!DOCTYPE html>
<html>
<head>
<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
<title>HTML5 Physics simulation</title>
<script type="text/x-mathjax-config">
MathJax.Hub.Config({ tex2jax: {inlineMath: [['$','$'], ['\\(','\\)']]} });
</script>
<script type="text/javascript"
src="https://cdn.rawgit.com/mathjax/MathJax/2.7.1/MathJax.js?config=TeX-AMS-MML_HTMLorMML">
</script>
<script>
(function(i,s,o,g,r,a,m){i['GoogleAnalyticsObject']=r;i[r]=i[r]||function(){
(i[r].q=i[r].q||[]).push(arguments)},i[r].l=1*new Date();a=s.createElement(o),
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})(window,document,'script','https://www.google-analytics.com/analytics.js','ga');
ga('create', 'UA-3425939-7', 'auto');
ga('send', 'pageview');
</script>
<style type="text/css">
<!--
body { background-color:#ededed; font:norm2al 12px/18px Arial, Helvetica, sans-serif; }
h1 { display:block; width:600px; margin:20px auto; paddVing-bottom:20px; font:norm2al 24px/30px Georgia, "Times New Roman", Times, serif; color:#333; text-shadow: 1px 2px 3px #ccc; border-bottom:1px solid #cbcbcb; }
#container { width:600px; margin:0 auto; }
#myCanvas { background:#fff; border:1px solid #cbcbcb; }
#nav { display:block; width:100%; text-align:center; }
#nav li { display:block; font-weight:bold; line-height:21px; text-shadow:1px 1px 1px #fff; width:100px; height:21px; paddVing:5px; margin:0 10px; background:#e0e0e0; border:1px solid #ccc; -moz-border-radius:4px;-webkit-border-radius:4px; border-radius:4px; float:left; }
#nav li a { color:#000; display:block; text-decoration:none; width:100%; height:100%; }
-->
</style>
<script type="text/javascript">
var pi = 3.14159265359;
var console = []
//Vector class + math
function Vector(x,y) { this.x = x; this.y = y; }
function addV(a,b) { return new Vector(a.x+b.x, a.y+b.y); }
function subV(a,b) { return new Vector(a.x-b.x, a.y-b.y); }
function mulV(a,k) { return new Vector(a.x*k, a.y *k); }
function dotV(a,b) { return a.x*b.x+a.y*b.y; }
function crossKV(k,v) { return new Vector(-v.y*k, v.x *k); }
function crossVV(a,b) { return a.x*b.y-a.y*b.x; }
function norm2V(a) { return dotV(a,a); }
function lengthV(a) { return Math.sqrt(norm2V(a,a)); }
function norm2(a) { return mulV(a, 1.0/lengthV(a)); }
function RotV(a, v) { return new Vector(Math.cos(a)*v.x - Math.sin(a)*v.y, Math.sin(a)*v.x+Math.cos(a)*v.y); }
function InvRotV(a, v) { return new Vector(Math.cos(a)*v.x + Math.sin(a)*v.y, -Math.sin(a)*v.x+Math.cos(a)*v.y); }
// Frame of reference
function Frame(x, y, r) { this.x = x; this.y = y; this.r = r; }
function FrameV(v, r) { this.x = v.x; this.y = v.y; this.r = r; }
function addF(a,b) { return new Frame(a.x+b.x, a.y+b.y, a.r+b.r); }
function subF(a,b) { return new Frame(a.x-b.x, a.y-b.y, a.r-b.r); }
function mulF(a,k) { return new Frame(a.x*k, a.y*k, a.r*k); }
function dotF(a,b) { return a.x*b.x+a.y*b.y+a.r*b.r; }
function TransFV(f,p) { return addV(new Vector(f.x, f.y), RotV(f.r, p) ); }
function InvTransFV(f,p) { return addV(InvRotV(f.r, p), new Vector(f.x, f.y) ); }
function getPosF(f) { return new Vector(f.x, f.y); }
// velocities, positions and radius
function State(p, v, b, M, I) { this.p = p; this.v = v; this.b = b; this.M=M; this.I = I; }
function getVelS(f, p) { return addV( f.v, crossKV( f.v.r, subV(p,f.p) ) ); }
function getPosS(f, p) { return TransFV( f.p, p ); }
var States = [];
var intergrationStep = 1./240;
//drawing helpers
function drawSphere(context, p, radius, color)
{
context.beginPath();
context.fillStyle=color;
context.arc(p.x,p.y,radius,0,Math.PI*2,true);
context.closePath();
context.fill();
}
function drawCross(context, p, radius, color)
{
drawLine( addV(p, new Vector(-1/5,0)),addV(p, new Vector(1/5,0)));
drawLine( addV(p, new Vector(0,-1/5)),addV(p, new Vector(0,1/5)));
}
function drawPoly(context, state, color)
{
for(var i=0;i<state.b.length-1;i++)
{
var p1 = TransFV(state.p, state.b[i]);
var p2 = TransFV(state.p, state.b[i+1]);
drawLine(p1,p2);
}
var p1 = TransFV(state.p, state.b[state.b.length-1]);
var p2 = TransFV(state.p, state.b[0]);
drawLine(p1,p2);
}
function drawLine( p1,p2)
{
context.beginPath();
context.fillStyle="#ff0000";
x = p1.x*30;
y = p1.y*30;
context.moveTo(x,y);
x = p2.x*30;
y = p2.y*30;
context.lineTo(x,y);
context.stroke();
}
function IsPointAboveRectEx(c, p, a, b)
{
var ab = subV(b,a);
var n = new Vector(ab.y, -ab.x);
var ap = subV(p-a);
var depth = DotV(n,ap);
if (depth<0)
{
var tt = DotV(ab,ap);
if (tt>0)
{
ln = lengthV(n);
if (tt/ln<ln)
{
depth /=ln;
return [depth, mulV(n,1.0/ln)];
}
}
}
return null;
}
function IsPointAboveRect(p, a, b)
{
var ab = subV(b,a);
var n = new Vector(ab.y, -ab.x);
var ap = subV(p-a);
var depth = DotV(n,ap);
return depth;
}
function IsPointInsideBody(p, b)
{
var out = 0;
var res;
for(var j=0;j<b.length;j++)
{
res = IsPointAboveRect(p, b[i], b[i+1]);
if (res==null)
out++;
}
res = IsPointAboveRect(p, b[b.length-1], b[0]);
if (res==null)
out++;
if (out==b.length) //is out
return;
}
function RectRectCollision(s1, s2)
{
var br1 = [];
for(var i=0;i<s1.b.length;i++)
{
br1[i] = TransFV(s1.p,s1.b[i]);
}
var br2 = [];
for(var i=0;i<s2.b.length;i++)
{
br2[i] = TransFV(s2.p,s2.b[i]);
}
//hereeeeeeeeeeeeeeee
for(var i=0;i<s1.length;i++)
{
var out = 0;
for(var j=0;j<s2.length;j++)
{
var depth = IsPointAboveRect(s2.b[j], s1.b[i], br1[i+1]);
if (depth<=0)
out+=1<<j;
}
var depth = IsPointAboveRect(br2[j], br1[br2.length-1], br1[0]);
if (out==0 && depth>0) //is out
return;
}
}
//
//multiply 2 matrices made out of vectors
//
function MulMatFF(m1, m2)
{
var O = [];
for(var j=0;j<m1.length;j++)
{
O[j]=[];
for(var i=0;i<m2[0].length;i++)
{
var tmp=0;
for(var k=0;k<m1[i].length;k++)
{
tmp += dotF(m1[j][k], m2[k][i]);
//tmp += dotV(m1[j][k], m2[k][i]);
//tmp += (m1[j][k] * m2[k][i]);
}
O[j][i] = tmp;
}
}
return O;
}
//
//multiply 2 matrices made out of vectors
//
function MulMatSF(s, f)
{
var O = [];
for(var j=0;j<f.length;j++)
{
O[j]=[];
for(var i=0;i<f[0].length;i++)
{
O[j][i] = new FrameV( mulV(f[j][i], s[j].M), f[j][i].r*s[j].I);
}
}
return O;
}
//
// multiply 2 matrices made out of scalars
//
function MulMatKK(m1, m2)
{
var O = [];
for(var j=0;j<m1.length;j++)
{
O[j]=[];
for(var i=0;i<m2[0].length;i++)
{
var tmp=0;
for(var k=0;k<m1[i].length;k++)
{
tmp += (m1[j][k] * m2[k][i]);
}
O[j][i] = tmp;
}
}
return O;
}
//
// multiply 2 matrices, one made out of vectors and the other made out of scalars
//
function MulMatFK(m1, m2)
{
var O = [];
for(var j=0;j<m1.length;j++)
{
O[j]=[];
for(var i=0;i<m2[0].length;i++)
{
var tmp= new Frame(0,0,0);
for(var k=0;k<m1[i].length;k++)
{
tmp = addF(mulF(m1[j][k], m2[k][i]), tmp);
//tmp = addV(mulV(m1[j][k], m2[k][i]), tmp);
}
O[j][i] = tmp;
}
}
return O;
}
//
// addV 2 matrices made out of scalars
//
function addVMatKK(m1, m2)
{
var O = [];
for(var j=0;j<m1.length;j++)
{
O[j]=[];
for(var i=0;i<m1[0].length;i++)
{
O[j][i] = m1[j][i] + m2[j][i];
}
}
return O;
}
//
// Transpose matrix
//
function TransposeMat(m)
{
var O = [];
for(var j=0;j<m[0].length;j++)
{
O[j]=[];
for(var i=0;i<m.length;i++)
{
O[j][i] = m[i][j];
}
}
return O;
}
//
// Get Jtranspose * lambda
//
function GetJtlambda(J, Minv, velT, bias)
{
var JvelT = MulMatFF(J, velT);
if (bias!=undefined)
{
JvelT = addVMatKK(JvelT, bias);
}
var Jt = TransposeMat(J);
var MinvJt = MulMatSF(Minv, Jt);
var JJt = MulMatFF(J, MinvJt );
var invJJt = invertMat( JJt );
var lambda = MulMatKK( invJJt, JvelT);
var Jtlambda = MulMatFK( Jt, lambda);
return Jtlambda;
}
//
// N particle pendulum, using a loop to generate J.
// Inputs:
// bodies: list of particle index
// top: coordinates of the top of the chain
//
function NPendulum(TopChain, bodies)
{
//compute jacobian
var J = []
for(var j=0;j<bodies.length;j++)
{
J[j] = [];
for(var i=0;i<bodies.length;i++)
{
if (i==0 && j==0)
J[j].push(subV(States.p[bodies[i]], TopChain));
else if (i==j)
J[j].push(subV(States.p[bodies[i]], States.p[bodies[i-1]]));
else if (i==j-1)
J[j].push(subV(States.p[bodies[j-1]], States.p[bodies[j]]));
else
J[j].push(new Vector(0,0));
}
}
//compute vels
var vel = [[]];
for(var i=0;i<bodies.length;i++)
vel[0].push(States.v[bodies[i]]);
var velT = TransposeMat(vel);
//compute bias
var distance = 10*10+10*10;
var betaoverh = .2/intergrationStep;
var bias = [[]];
bias[0].push( betaoverh * .5*(norm2V(subV( States.p[bodies[0]], TopChain)) - distance));
for(var i=1;i<bodies.length;i++)
{
bias[0].push( betaoverh * .5 * (norm2V(subV( States.p[bodies[i-1]], States.p[bodies[i]])) - distance));
}
var biasT = TransposeMat(bias);
//solver
var Jtlambda = GetJtlambda(J, velT, biasT);
//update speeds
for(var i=0;i<bodies.length;i++)
States.v[bodies[i]] = subV(States.v[bodies[i]], Jtlambda[i][0]);
//draw connections
drawLine( TopChain, States.p[bodies[0]])
for(var i=1;i<bodies.length;i++)
drawLine( States.p[bodies[i-1]], States.p[bodies[i]]);
}
//
// Distance constraint, note how the jacobian is a small matrix
// Inputs:
// bodyA, bodyB: particle index
//
function DistanceConstraint(bodyA, Pa, bodyB, Pb)
{
//compute jacobian
var J = []
var PaPb = subV(Pb, Pa);
var PbPa = subV(Pa, Pb);
var Ca = getPosF(States[bodyA].p);
var Cb = getPosF(States[bodyB].p);
var CaPa = subV(Pa, Ca);
var CbPb = subV(Pb, Cb);
var Wa = -crossVV(CaPa, PaPb);
var Wb = crossVV(CbPb, PaPb);
J[0] = [ new FrameV(mulV(mulV(PaPb,-1),2), 2*Wa), new FrameV(mulV(PaPb,2), 2*Wb) ];
//document.getElementById("text").innerHTML += "Wa " + (J[0][0].r).toFixed(2) + " " + (J[0][1].r).toFixed(2) + "<br>";
//compute vels
var vel = [[ States[bodyA].v, States[bodyB].v ]];
var velT = TransposeMat(vel);
//compute bias
var betaoverh = .5/intergrationStep;
var C = dotV(PaPb, PaPb);
if (C<= 1e-15)
return;
var bias = [[betaoverh * C]];
//just the diagonal matrix
var Minv = [ States[bodyA], States[bodyB] ];
//solver
var Jtlambda = GetJtlambda(J, Minv, velT, bias)
var MinvJtlambda = MulMatSF(Minv, Jtlambda);
//update speeds
States[bodyA].v = subF(States[bodyA].v, MinvJtlambda[0][0]);
States[bodyB].v = subF(States[bodyB].v, MinvJtlambda[1][0]);
//document.getElementById("text").innerHTML += "Sa " + States[bodyA].v.x + " " + States[bodyA].v.y + " "+ States[bodyA].v.r + "<br>";
//document.getElementById("text").innerHTML += "Sb " + States[bodyB].v.x + " " + States[bodyB].v.y + " "+ States[bodyB].v.r + "<br>";
}
//
// Distance constraint, note how the jacobian is a small matrix
// Inputs:
// bodyA, bodyB: particle index
//
function ContactConstraintBallBall(bodyA, norm2al, bodyB)
{
var PaCa = subV(contact.pA, getPosF(States[contact.bodyA].p));
var PbCb = subV(contact.pB, getPosF(States[contact.bodyB].p));
//compute jacobian
var J = []
J[0] = [
new Frame( contact.N.x, contact.N.y, crossVV(PaCa, contact.N)),
new Frame( -contact.N.x, -contact.N.y, -crossVV(PbCb, contact.N))
];
//compute vels
var vel = [[ States[bodyA].v, States[bodyB].v ]];
var velT = TransposeMat(vel);
//compute bias
var betaoverh = .1/intergrationStep;
var C = dotV(subV(States[bodyA].p, States[bodyB].p), norm2al) - (r[bodyA]+r[bodyB]);
//bouncy term
var vn = 0;//dotV(subV(States[bodyA].v, States[bodyB].v), norm2al);
var bias = [[betaoverh * C + vn]];
//solver
var Jtlambda = GetJtlambda(J, velT, bias)
//update speeds
States[contact.bodyA].v = subF(States[contact.bodyA].v, Jtlambda[0][0]);
States[contact.bodyB].v = subF(States[contact.bodyB].v, Jtlambda[1][0]);
}
function ContactConstraintBallND(contact)
{
var Ca = getPosF(States[contact.bodyA].p);
var PaCa = subV(contact.pA, Ca);
var w = crossVV(PaCa, contact.N);
//compute jacobian
var J = []
J[0] = [
new Frame( contact.N.x, contact.N.y, w)
];
//compute vels
var vel = [[ States[contact.bodyA].v ]];
var velT = TransposeMat(vel);
//compute bias
var betaoverh = .8/intergrationStep;
var C = -contact.depth;
//bouncy term
var vn = 0;//-0.9*dotV(States[contact.bodyA].fv.p, contact.N);
var bias = [[betaoverh * C + vn]];
//solver
var Jtlambda = GetJtlambda(J, velT, bias)
//update speeds
States[contact.bodyA].v = subF(States[contact.bodyA].v, Jtlambda[0][0]);
}
//
// Integrate step
//
function integrate(t)
{
for(var i=0;i<States.length;i++)
{
States[i].p = addF(States[i].p ,mulF(States[i].v, t));
//States[i].v.y += .5;
//States.v[i] = mulV(States.v[i], .98);
}
}
function Collision(bodyA, pA, bodyB, pB, N, depth)
{
this.bodyA = bodyA;
this.bodyB = bodyB;
this.pA = pA;
this.pB = pB;
this.N = N;
this.depth= depth;
}
function ComputeCollsions()
{
var colliders= [];
/*
//compute collision ball/ball
for(var i=0;i<States.length;i++)
{
for(var j=0;j<i;j++)
{
var rr = (r[i]+r[j]);
var diff = subV(States[i].p, States[j].p);
var d = dotV(diff, diff);
if (d<rr*rr)
{
colliders.pA.push(i);
colliders.pB.push(j);
colliders.N.push( mulV(diff,1.0/Math.sqrt(d) ) );
}
}
}
// collisions ball/plane
for(var i=0;i<States.length;i++)
{
if (States[i].p.y+r[i]>300)
{
colliders.push( new Collision(pA, undefined, new Vector(0,-1) );
}
}
*/
// collisions box/plane
for(var i=0;i<States.length;i++)
{
var b = States[i].b;
for(var v=0;v<b.length;v++)
{
var p = TransFV(States[i].p, b[v]);
var depth = p.y-300;
if (depth>0)
{
colliders.push( new Collision(i, p, undefined, undefined, new Vector(0,-1), depth) );
}
}
}
return colliders;
}
function draw()
{
//apply constraints
var iterations = 4;
for(var iter=0;iter<iterations;iter++)
{
for(var i=0;i<States.length-1;i++)
{
var Pa = getPosS( States[i], States[i].b[3]);
var Pb = getPosS( States[i+1], States[i+1].b[0]);
DistanceConstraint(i, Pa, i+1, Pb)
}
}
//render chain
{
context= myCanvas.getContext('2d');
context.clearRect(0,0,600,600);
for(var i=0;i<States.length;i++)
{
drawPoly(context, States[i], "#ff0000");
}
}
//apply gravity
for(var i=1;i<States.length;i++)
States[i].v.y+=9.8/10;
//compute tentative velocities
integrate(intergrationStep);
}
function init()
{
var cubeVerts = [
new Vector(-1,-1),
new Vector(1,-1),
new Vector(1,1),
new Vector(0, 1),
new Vector(-1,1)
];
var linkVerts = [ new Vector(0, -0.5), new Vector(-0.1, -0.3), new Vector(-0.1, 0.3), new Vector(0, 0.5), new Vector(0.1, 0.3), new Vector(0.1, -0.3)];
var p = new Frame(10,0, 0*pi/180);
var v = new Frame(0,0, 0);
States.push( new State(p, v, cubeVerts, 0, 0) );
for(var i=0;i<15;i++)
{
var l = Math.sqrt(.5)
var p = new Frame(10+i*l + l*.5,1+i*l + l*.5, -45*pi/180);
var v = new Frame(0,0, 0);
States.push( new State(p, v, linkVerts, 5, 5) );
}
setInterval(draw,10);
}
//
// Inverts a matrix (taken from http://blog.acipo.com/matrix-inversion-in-javascript/)
//
function invertMat(M){
// I use Guassian Elimination to calculate the inverse:
// (1) 'augment' the matrix (left) by the identity (on the right)
// (2) Turn the matrix on the left into the identity by elemetry row ops
// (3) The matrix on the right is the inverse (was the identity matrix)
// There are 3 elemtary row ops: (I combine b and c in my code)
// (a) Swap 2 rows
// (b) Multiply a row by a scalar
// (c) addV 2 rows
//if the matrix isn't square: exit (error)
if(M.length !== M[0].length){return;}
//create the identity matrix (I), and a copy (C) of the original
var i=0, ii=0, j=0, dim=M.length, e=0, t=0;
var I = [], C = [];
for(i=0; i<dim; i+=1){
// Create the row
I[I.length]=[];
C[C.length]=[];
for(j=0; j<dim; j+=1){
//if we're on the diagonal, put a 1 (for identity)
if(i==j){ I[i][j] = 1; }
else{ I[i][j] = 0; }
// Also, make the copy of the original
C[i][j] = M[i][j];
}
}
// Perform elementary row operations
for(i=0; i<dim; i+=1){
// get the element e on the diagonal
e = C[i][i];
// if we have a 0 on the diagonal (we'll need to swap with a lower row)
if(e==0){
//look through every row below the i'th row
for(ii=i+1; ii<dim; ii+=1){
//if the ii'th row has a non-0 in the i'th col
if(C[ii][i] != 0){
//it would make the diagonal have a non-0 so swap it
for(j=0; j<dim; j++){
e = C[i][j]; //temp store i'th row
C[i][j] = C[ii][j];//replace i'th row by ii'th
C[ii][j] = e; //repace ii'th by temp
e = I[i][j]; //temp store i'th row
I[i][j] = I[ii][j];//replace i'th row by ii'th
I[ii][j] = e; //repace ii'th by temp
}
//don't bother checking other rows since we've swapped
break;
}
}
//get the new diagonal
e = C[i][i];
//if it's still 0, not invertable (error)
if(e==0){return}
}
// Scale this row down by e (so we have a 1 on the diagonal)
for(j=0; j<dim; j++){
C[i][j] = C[i][j]/e; //apply to original matrix
I[i][j] = I[i][j]/e; //apply to identity
}
// subVtract this row (scaled appropriately for each row) from ALL of
// the other rows so that there will be 0's in this column in the
// rows above and below this one
for(ii=0; ii<dim; ii++){
// Only apply to other rows (we want a 1 on the diagonal)
if(ii==i){continue;}
// We want to change this element to 0
e = C[ii][i];
// subVtract (the row above(or below) scaled by e) from (the
// current row) but start at the i'th column and assume all the
// stuff left of diagonal is 0 (which it should be if we made this
// algorithm correctly)
for(j=0; j<dim; j++){
C[ii][j] -= e*C[i][j]; //apply to original matrix
I[ii][j] -= e*I[i][j]; //apply to identity
}
}
}
//we've done all operations, C should be the identity
//matrix I should be the inverse:
return I;
}
</script>
</head>
<body onload="init()">
<h1>Rigid solid chain (sequential solver)</h1>
<div id="container">
<canvas id="myCanvas" width="600" height="600"></canvas>
<div id="text"></div>
<h2>Intro</h2>
This is a quick exercise to learn how constraints work in a physics simulator. This sample is using equality constraints for the joints</br>
</br>
This physics simulator is heavily based on Erin Catto's GDC2009 talk.</br>
</br>
</br>
<h2>Contact/Questions:</h2>
<my_github_account_username>$@gmail.com$.
</br>
</br>
</div>
</body>
</html>