mathjs
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Math.js is an extensive math library for JavaScript and Node.js. It features a flexible expression parser with support for symbolic computation, comes with a large set of built-in functions and constants, and offers an integrated solution to work with dif
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<title>math.js | rocket trajectory optimization</title>
<script src="../../dist/math.js"></script>
<script src="https://cdnjs.cloudflare.com/ajax/libs/Chart.js/2.5.0/Chart.min.js"></script>
<style>
body {
font-family: sans-serif;
}
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<h1>Rocket trajectory optimization</h1>
<p>
This example simulates the ascent stage of the Apollo Lunar Module modeled using a system of ordinary differential equations.
</p>
<canvas id=canvas1 width=1600 height=400></canvas>
<canvas id=canvas2 width=800 height=400></canvas>
<script>
function ndsolve(f, x0, dt, tmax) {
var n = f.size()[0]; // Number of variables
var x = x0.clone(); // Current values of variables
var dxdt = []; // Temporary variable to hold time-derivatives
var result = []; // Contains entire solution
var nsteps = math.divide(tmax, dt); // Number of time steps
for(var i=0; i<nsteps; i++) {
// Compute derivatives
for(var j=0; j<n; j++) {
dxdt[j] = f.get([j]).apply(null, x.toArray());
}
// Euler method to compute next time step
for(var j=0; j<n; j++) {
x.set([j], math.add(x.get([j]), math.multiply(dxdt[j], dt)));
}
result.push(x.clone());
}
return math.matrix(result);
}
// Import the numerical ODE solver
math.import({ndsolve:ndsolve});
// Create a math.js context for our simulation. Everything else occurs in the context of the expression parser!
var sim = math.parser();
sim.eval("G = 6.67408e-11 m^3 kg^-1 s^-2"); // Gravitational constant
sim.eval("mbody = 5.972e24 kg"); // Mass of Earth
sim.eval("mu = G * mbody");
sim.eval("dt = 1.0 s"); // Simulation timestep
sim.eval("tfinal = 162 s"); // Simulation duration
sim.eval("T = 1710000 lbf * 0.9"); // Engine thrust
sim.eval("g0 = 9.80665 m/s^2"); // Standard gravity: used for calculating prop consumption (dmdt)
sim.eval("isp = 290 s"); // Specific impulse
sim.eval("gamma0 = 89.99883 deg"); // Initial pitch angle (90 deg is vertical)
sim.eval("r0 = 6378.1370 km"); // Equatorial radius of Earth
sim.eval("v0 = 10 m/s"); // Initial velocity (must be non-zero because ODE is ill-conditioned)
sim.eval("phi0 = 0 deg"); // Initial orbital reference angle
sim.eval("m0 = 1207920 lbm + 30000 lbm"); // Initial mass of rocket and fuel
// Define the equations of motion. It is important to maintain the same argument order for each of these functions.
sim.eval("drdt(r, v, m, phi, gamma) = v sin(gamma)");
sim.eval("dvdt(r, v, m, phi, gamma) = -mu / r^2 sin(gamma) + T / m");
sim.eval("dmdt(r, v, m, phi, gamma) = -T/g0/isp");
sim.eval("dphidt(r, v, m, phi, gamma) = v/r cos(gamma) * rad");
sim.eval("dgammadt(r, v, m, phi, gamma) = (1/r * (v - mu / (r v)) * cos(gamma)) * rad");
// Again, remember to maintain the same variable order in the call to ndsolve.
sim.eval("result_stage1 = ndsolve([drdt, dvdt, dmdt, dphidt, dgammadt], [r0, v0, m0, phi0, gamma0], dt, tfinal)");
// Reset initial conditions for interstage flight
sim.eval("T = 0 lbf");
sim.eval("tfinal = 12 s");
sim.eval("x = flatten(result_stage1[result_stage1.size()[1],:])");
sim.eval("result_interstage = ndsolve([drdt, dvdt, dmdt, dphidt, dgammadt], x, dt, tfinal)");
console.log(sim.eval("result_interstage[result_interstage.size()[1],3]").toString());
// Reset initial conditions for stage 2 flight
sim.eval("T = 210000 lbf");
sim.eval("isp = 348 s");
sim.eval("tfinal = 397 s");
sim.eval("x = flatten(result_interstage[result_interstage.size()[1],:])");
sim.eval("x[3] = 273600 lbm"); // Lighten the rocket a bit since we discarded the first stage
sim.eval("result_stage2 = ndsolve([drdt, dvdt, dmdt, dphidt, dgammadt], x, dt, tfinal)");
// Reset initial conditions for unpowered flight
sim.eval("T = 0 lbf");
sim.eval("tfinal = 60 s");
sim.eval("x = flatten(result_stage2[result_stage2.size()[1],:])");
sim.eval("result_unpowered = ndsolve([drdt, dvdt, dmdt, dphidt, dgammadt], x, dt, tfinal)");
// Extract the useful information from the results so it can be plotted
var data_stage1 = sim.eval("concat( ( result_stage1[:,4]' - phi0) * r0 / rad / km, ( result_stage1[:,1]' - r0) / km, 1 )' ").toArray().map(function(e) { return {x: e[0], y: e[1]}; });
var data_interstage = sim.eval("concat( (result_interstage[:,4]' - phi0) * r0 / rad / km, (result_interstage[:,1]' - r0) / km, 1 )' ").toArray().map(function(e) { return {x: e[0], y: e[1]}; });
var data_stage2 = sim.eval("concat( ( result_stage2[:,4]' - phi0) * r0 / rad / km, ( result_stage2[:,1]' - r0) / km, 1 )' ").toArray().map(function(e) { return {x: e[0], y: e[1]}; });
var data_unpowered = sim.eval("concat( ( result_unpowered[:,4]' - phi0) * r0 / rad / km, ( result_unpowered[:,1]' - r0) / km, 1 )' ").toArray().map(function(e) { return {x: e[0], y: e[1]}; });
var chart = new Chart(document.getElementById('canvas1'), {
type: 'line',
data: {
datasets: [{
label: "Stage 1",
data: data_stage1,
fill: false,
borderColor: "red",
pointRadius: 0
}, {
label: "Interstage",
data: data_interstage,
fill: false,
borderColor: "green",
pointRadius: 0
}, {
label: "Stage 2",
data: data_stage2,
fill: false,
borderColor: "orange",
pointRadius: 0
}, {
label: "Unpowered",
data: data_unpowered,
fill: false,
borderColor: "blue",
pointRadius: 0
}]
},
options: {
scales: {
xAxes: [{
type: 'linear',
position: 'bottom'
}]
}
}
});
</script>
</body>
</html>