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<!DOCTYPE html>
<html>
<head>
<title>KalmanJS :: 2D Linear Kalman Filter</title>
<meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
<script src="lib/jquery/jquery-1.8.2.min.js"></script>
<script src="lib/jquery/jquery.slider-1.3.js"></script>
<link rel="stylesheet" href="lib/jquery/jquery.slider-1.3.css" type="text/css">
<script src="common.js"></script>
<script src="lib/plotjs/PlotJS.js"></script>
<script src="lib/sylvester/sylvester.js"></script>
<script src="lib/kalmanjs/LinearKalmanFilter.js"></script>
<link rel="stylesheet" href="style/style.css">
</head>
<body>
<canvas id="data-plot" class="plot-main"></canvas>
<canvas id="predicted-state-plot" class="plot-sub"></canvas>
<canvas id="predicted-probability-plot" class="plot-sub"></canvas>
<canvas id="covariance-plot" class="plot-sub"></canvas>
<canvas id="gain-plot" class="plot-sub"></canvas>
<canvas id="innovation-plot" class="plot-sub"></canvas>
<canvas id="innovation-covariance-plot" class="plot-sub"></canvas>
<div id="controls"></div>
<script>
// Multi-variable cannonball
Cannon = function(gravity, angle, muzzleVelocity, noise) {
this.angle = angle;
this.muzzleVelocity = muzzleVelocity;
this.gravity = [0, gravity];
this.velocity = [this.muzzleVelocity * Math.cos(this.angle * Math.PI / 180.0), this.muzzleVelocity * Math.sin(angle * Math.PI / 180.0)];
this.pos = [0, 0];
this.noise = noise;
this.getX = function() {
return this.pos[0];
};
this.getY = function() {
return this.pos[1];
};
this.getMeasurementX = function() {
return Math.randomGaussian(this.noise, this.getX());
};
this.getMeasurementY = function() {
return Math.randomGaussian(this.noise, this.getY());
};
this.getVelocityX = function() {
return this.velocity[0];
};
this.getVelocityY = function() {
return this.velocity[1];
};
this.hasLanded = function() {
return this.pos[1] < 0.0;
};
this.step = function(dt) {
var stepGravity = [
this.gravity[0] * dt,
this.gravity[1] * dt
];
this.velocity = [
this.velocity[0] + stepGravity[0],
this.velocity[1] + stepGravity[1]
];
var stepVelocity = [
this.velocity[0] * dt,
this.velocity[1] * dt
];
this.pos = [
this.pos[0] + stepVelocity[0],
this.pos[1] + stepVelocity[1]
];
};
};
var options = {
muzzleVelocity: 100.0,
angle: 45.0,
gravity: -9.81,
initialCovariance: 0.0,
processError: 0.0,
measurementError: 0.2,
noise: 30.0,
dt: 0.1,
observationCutoff: 1000
};
function run() {
var cannon = new Cannon(options.gravity, options.angle, options.muzzleVelocity, options.noise),
startPosX = 0.0,
startPosY = 0.0,
startVelocityX = options.muzzleVelocity * Math.cos(options.angle * Math.PI / 180),
startVelocityY = options.muzzleVelocity * Math.sin(options.angle * Math.PI / 180);
// This is the state transition vector, which represents part of the kinematics.
// 1, ts, 0, 0 => x(n+1) = x(n) + vx(n)
// 0, 1, 0, 0 => vx(n+1) = vx(n)
// 0, 0, 1, ts => y(n+1) = y(n) + vy(n)
// 0, 0, 0, 1 => vy(n+1) = vy(n)
// Remember, acceleration gets added to these at the control vector.
var stateTransitionMatrix = $M([
[1, options.dt, 0, 0],
[0, 1, 0, 0],
[0, 0, 1, options.dt],
[0, 0, 0, 1]
]);
// The control vector, which adds acceleration to the kinematic equations.
// 0 => x(n+1) = x(n+1)
// 0 => vx(n+1) = vx(n+1)
// -9.81*ts^2 => y(n+1) = y(n+1) + 0.5*-9.81*ts^2
// -9.81*ts => vy(n+1) = vy(n+1) + -9.81*ts
var controlMatrix = $M([
[0, 0, 0, 0],
[0, 0, 0, 0],
[0, 0, 1, 0],
[0, 0, 0, 1]
]),
controlVector = $M([
[0],
[0],
[0.5 * options.gravity * options.dt * options.dt],
[options.gravity * options.dt]
]);
// After state transition and control, here are the equations:
// x(n+1) = x(n) + vx(n)
// vx(n+1) = vx(n)
// y(n+1) = y(n) + vy(n) - 0.5*9.81*ts^2
// vy(n+1) = vy(n) + -9.81*ts
// Which, if you recall, are the equations of motion for a parabola. Perfect.
// Observation matrix is the identity matrix, since we can get direct
// measurements of all values in our example.
var observationMatrix = $M([
[1, 0, 0, 0],
[0, 1, 0, 0],
[0, 0, 1, 0],
[0, 0, 0, 1]
]);
// This is our guess of the initial state. I intentionally set the Y value
// wrong to illustrate how fast the Kalman filter will pick up on that.
var initialStateEstimate = $M([
[startPosX],
[startVelocityX],
[startPosY],
[startVelocityY]
]),
initialCovarianceEstimate = $M([
[options.initialCovariance, 0, 0, 0],
[0, options.initialCovariance, 0, 0],
[0, 0, options.initialCovariance, 0],
[0, 0, 0, options.initialCovariance]
]),
processErrorEstimate = $M([
[options.processError, 0, 0, 0],
[0, options.processError, 0, 0],
[0, 0, options.processError, 0],
[0, 0, 0, options.processError]
]),
measurementErrorEstimate = $M([
[options.measurementError, 0, 0, 0],
[0, options.measurementError, 0, 0],
[0, 0, options.measurementError, 0],
[0, 0, 0, options.measurementError]
]);
var filter = new LinearKalmanFilter(
stateTransitionMatrix,
controlMatrix,
observationMatrix,
initialStateEstimate,
initialCovarianceEstimate,
processErrorEstimate,
measurementErrorEstimate
),
dataPlot = new PlotJS(),
predictedStatePlot = new PlotJS(),
predictedProbabilityPlot = new PlotJS(),
covariancePlot = new PlotJS(),
gainPlot = new PlotJS(),
innovationPlot = new PlotJS(),
innovationCovariancePlot = new PlotJS(),
realPos = {},
measuredPos = {},
filterPos = {},
predictedPosX = [], predictedPosY = [], predictedVelX = [], predictedVelY = [],
probabilityPosX = [], probabilityPosY = [], probabilityVelX = [], probabilityVelY = [],
covariancePosX = [], covariancePosY = [], covarianceVelX = [], covarianceVelY = [],
gainPosX = [], gainPosY = [], gainVelX = [], gainVelY = [],
innovationPosX = [], innovationPosY = [], innovationVelX = [], innovationVelY = [],
innovationCovariancePosX = [], innovationCovariancePosY = [], innovationCovarianceVelX = [], innovationCovarianceVelY = [],
realX, realY,
measurementX, measurementY,
velocityX, velocityY,
measurementVector,
filterState, filterX, filterY,
gain,
innovation,
innovationCovariance,
step = 0;
while (!cannon.hasLanded()) {
cannon.step(options.dt);
realX = cannon.getX();
realY = cannon.getY();
measurementX = cannon.getMeasurementX();
measurementY = cannon.getMeasurementY();
velocityX = cannon.getVelocityX();
velocityY = cannon.getVelocityY();
measurementVector = $M([
[measurementX],
[velocityX],
[measurementY],
[velocityY]
]);
filter.predict(controlVector);
filter.observe(realX < options.observationCutoff ? measurementVector : null);
filter.update();
filterState = filter.getStateEstimate();
/*q = {
x: covariance.a11
}*/
filterX = filterState.e(1, 1);
filterY = filterState.e(3, 1);
realPos[realX] = realY;
measuredPos[measurementX] = measurementY;
filterPos[filterX] = filterY;
predictedPosY.push(filter.predictedStateEstimate.e(3, 1));
predictedPosX.push(filter.predictedStateEstimate.e(1, 1));
predictedVelY.push(filter.predictedStateEstimate.e(4, 1));
predictedVelX.push(filter.predictedStateEstimate.e(2, 1));
probabilityPosY.push(filter.predictedProbabilityEstimate.e(3, 1));
probabilityPosX.push(filter.predictedProbabilityEstimate.e(1, 1));
probabilityVelY.push(filter.predictedProbabilityEstimate.e(4, 1));
probabilityVelX.push(filter.predictedProbabilityEstimate.e(2, 1));
covariancePosY.push(filter.covarianceEstimate.e(3, 1));
covariancePosX.push(filter.covarianceEstimate.e(1, 1));
covarianceVelY.push(filter.covarianceEstimate.e(4, 1));
covarianceVelX.push(filter.covarianceEstimate.e(2, 1));
gainPosY.push(filter.kalmanGain.e(3, 1));
gainPosX.push(filter.kalmanGain.e(1, 1));
gainVelY.push(filter.kalmanGain.e(4, 1));
gainVelX.push(filter.kalmanGain.e(2, 1));
innovationPosY.push(filter.innovation.e(3, 1));
innovationPosX.push(filter.innovation.e(1, 1));
innovationVelY.push(filter.innovation.e(4, 1));
innovationVelX.push(filter.innovation.e(2, 1));
innovationCovariancePosY.push(filter.innovationCovariance.e(3, 1));
innovationCovariancePosX.push(filter.innovationCovariance.e(1, 1));
innovationCovarianceVelY.push(filter.innovationCovariance.e(4, 1));
innovationCovarianceVelX.push(filter.innovationCovariance.e(2, 1));
step++;
//console.log('#' + step, filter);
}
dataPlot
.addData(realPos, 'Real position')
.addData(measuredPos, 'Measured position')
.addData(filterPos, 'Filter position')
.setTitle('Multi-Variable Kalman Filter - Cannonball Flight')
.draw('data-plot', {
showPoints: false,
drawLegend: true,
paddingTop: 60,
rangeMin: 0,
rangeMax: 100
});
predictedStatePlot
.addData(predictedPosX, 'Px')
.addData(predictedPosY, 'Py')
.addData(predictedVelX, 'Vx')
.addData(predictedVelY, 'Vy')
.setTitle('State estimate')
.draw('predicted-state-plot', {
showPoints: false,
drawLegend: true,
valueDecimals: 1,
paddingTop: 60,
rangeMin: 0,
rangeMax: 250
});
predictedProbabilityPlot
.addData(probabilityPosX, 'Px')
.addData(probabilityPosY, 'Py')
.addData(probabilityVelX, 'Vx')
.addData(probabilityVelY, 'Vy')
.setTitle('Probability estimate')
.draw('predicted-probability-plot', {
showPoints: false,
drawLegend: true,
valueDecimals: 1,
paddingTop: 60,
rangeMin: 0,
rangeMax: 5
});
covariancePlot
.addData(covariancePosX, 'Px')
.addData(covariancePosY, 'Py')
.addData(covarianceVelX, 'Vx')
.addData(covarianceVelY, 'Vy')
.setTitle('Covariance estimate')
.draw('covariance-plot', {
showPoints: false,
drawLegend: true,
valueDecimals: 1,
paddingTop: 60,
rangeMin: 0,
rangeMax: 5
});
gainPlot
.addData(gainPosX, 'Px')
.addData(gainPosY, 'Py')
.addData(gainVelX, 'Vx')
.addData(gainVelY, 'Vy')
.setTitle('Kalman gain')
.draw('gain-plot', {
showPoints: false,
drawLegend: true,
valueDecimals: 1,
paddingTop: 60,
rangeMin: 0,
rangeMax: 0.5
});
innovationPlot
.addData(innovationPosX, 'Px')
.addData(innovationPosY, 'Py')
.addData(innovationVelX, 'Vx')
.addData(innovationVelY, 'Vy')
.setTitle('Innovation')
.draw('innovation-plot', {
showPoints: false,
drawLegend: true,
valueDecimals: 1,
paddingTop: 60,
rangeMin: -30,
rangeMax: 30
});
innovationCovariancePlot
.addData(innovationCovariancePosX, 'Px')
.addData(innovationCovariancePosY, 'Py')
.addData(innovationCovarianceVelX, 'Vx')
.addData(innovationCovarianceVelY, 'Vy')
.setTitle('Innovation covariance')
.draw('innovation-covariance-plot', {
showPoints: false,
drawLegend: true,
valueDecimals: 1,
paddingTop: 60,
rangeMin: -1,
rangeMax: 1
});
}
var controls = [
// control value min max step
['muzzleVelocity', 50.0, 0, 100, 1],
['angle', 60.0, 1, 90, 1],
['gravity', -9.81, -20, 0, 0.1],
['initialCovariance', 50.0, 0, 100, 1],
['processError', 0.0, 0, 1, 0.001],
['measurementError', 50, 0, 100, 0.1],
['noise', 10.0, 0, 50, 1],
['dt', 0.02, 0.01, 0.2, 0.01],
['observationCutoff', 250, 10, 500, 10]
];
for (var i = 0; i < controls.length; i++) {
createControl.apply(this, controls[i]);
}
run();
</script>
<div id="github"><a href="https://github.com/kallaspriit/KalmanJS" target="_blank">Github Project</a><a href="https://github.com/kallaspriit/KalmanJS/zipball/master">Download Latest</a></div>
</body>
</html>