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153 lines
6.3 KiB
HTML
153 lines
6.3 KiB
HTML
<!DOCTYPE html>
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<html lang="en">
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<head>
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<meta charset="utf-8">
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<title>math.js | rocket trajectory optimization</title>
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<script src="../../dist/math.js"></script>
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<script src="https://cdnjs.cloudflare.com/ajax/libs/Chart.js/2.5.0/Chart.min.js"></script>
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<style>
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body {
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font-family: sans-serif;
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}
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</style>
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</head>
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<body>
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<h1>Rocket trajectory optimization</h1>
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<p>
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This example simulates the ascent stage of the Apollo Lunar Module modeled using a system of ordinary differential equations.
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</p>
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<canvas id=canvas1 width=1600 height=400></canvas>
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<canvas id=canvas2 width=800 height=400></canvas>
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<script>
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function ndsolve(f, x0, dt, tmax) {
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const n = f.size()[0] // Number of variables
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const x = x0.clone() // Current values of variables
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const dxdt = [] // Temporary variable to hold time-derivatives
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const result = [] // Contains entire solution
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const nsteps = math.divide(tmax, dt) // Number of time steps
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for(let i=0; i<nsteps; i++) {
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// Compute derivatives
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for(let j=0; j<n; j++) {
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dxdt[j] = f.get([j]).apply(null, x.toArray())
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}
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// Euler method to compute next time step
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for(let j=0; j<n; j++) {
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x.set([j], math.add(x.get([j]), math.multiply(dxdt[j], dt)))
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}
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result.push(x.clone())
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}
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return math.matrix(result)
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}
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// Import the numerical ODE solver
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math.import({ndsolve:ndsolve})
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// Create a math.js context for our simulation. Everything else occurs in the context of the expression parser!
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const sim = math.parser()
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sim.evaluate("G = 6.67408e-11 m^3 kg^-1 s^-2") // Gravitational constant
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sim.evaluate("mbody = 5.972e24 kg") // Mass of Earth
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sim.evaluate("mu = G * mbody")
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sim.evaluate("dt = 1.0 s") // Simulation timestep
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sim.evaluate("tfinal = 162 s") // Simulation duration
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sim.evaluate("T = 1710000 lbf * 0.9") // Engine thrust
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sim.evaluate("g0 = 9.80665 m/s^2") // Standard gravity: used for calculating prop consumption (dmdt)
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sim.evaluate("isp = 290 s") // Specific impulse
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sim.evaluate("gamma0 = 89.99883 deg") // Initial pitch angle (90 deg is vertical)
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sim.evaluate("r0 = 6378.1370 km") // Equatorial radius of Earth
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sim.evaluate("v0 = 10 m/s") // Initial velocity (must be non-zero because ODE is ill-conditioned)
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sim.evaluate("phi0 = 0 deg") // Initial orbital reference angle
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sim.evaluate("m0 = 1207920 lbm + 30000 lbm") // Initial mass of rocket and fuel
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// Define the equations of motion. It is important to maintain the same argument order for each of these functions.
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sim.evaluate("drdt(r, v, m, phi, gamma) = v sin(gamma)")
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sim.evaluate("dvdt(r, v, m, phi, gamma) = -mu / r^2 * sin(gamma) + T / m")
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sim.evaluate("dmdt(r, v, m, phi, gamma) = -T/g0/isp")
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sim.evaluate("dphidt(r, v, m, phi, gamma) = v/r * cos(gamma) * rad")
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sim.evaluate("dgammadt(r, v, m, phi, gamma) = (1/r * (v - mu / (r v)) * cos(gamma)) * rad")
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// Again, remember to maintain the same variable order in the call to ndsolve.
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sim.evaluate("result_stage1 = ndsolve([drdt, dvdt, dmdt, dphidt, dgammadt], [r0, v0, m0, phi0, gamma0], dt, tfinal)")
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// Reset initial conditions for interstage flight
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sim.evaluate("T = 0 lbf")
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sim.evaluate("tfinal = 12 s")
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sim.evaluate("x = flatten(result_stage1[result_stage1.size()[1],:])")
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sim.evaluate("result_interstage = ndsolve([drdt, dvdt, dmdt, dphidt, dgammadt], x, dt, tfinal)")
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console.log(sim.evaluate("result_interstage[result_interstage.size()[1],3]").toString())
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// Reset initial conditions for stage 2 flight
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sim.evaluate("T = 210000 lbf")
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sim.evaluate("isp = 348 s")
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sim.evaluate("tfinal = 397 s")
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sim.evaluate("x = flatten(result_interstage[result_interstage.size()[1],:])")
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sim.evaluate("x[3] = 273600 lbm") // Lighten the rocket a bit since we discarded the first stage
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sim.evaluate("result_stage2 = ndsolve([drdt, dvdt, dmdt, dphidt, dgammadt], x, dt, tfinal)")
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// Reset initial conditions for unpowered flight
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sim.evaluate("T = 0 lbf")
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sim.evaluate("tfinal = 60 s")
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sim.evaluate("x = flatten(result_stage2[result_stage2.size()[1],:])")
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sim.evaluate("result_unpowered = ndsolve([drdt, dvdt, dmdt, dphidt, dgammadt], x, dt, tfinal)")
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// Extract the useful information from the results so it can be plotted
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const data_stage1 = sim.evaluate("transpose(concat( transpose( result_stage1[:,4] - phi0) * r0 / rad / km, ( transpose(result_stage1[:,1]) - r0) / km, 1 ))").toArray().map(function(e) { return {x: e[0], y: e[1]} })
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const data_interstage = sim.evaluate("transpose(concat( transpose(result_interstage[:,4] - phi0) * r0 / rad / km, (transpose(result_interstage[:,1]) - r0) / km, 1 ))").toArray().map(function(e) { return {x: e[0], y: e[1]} })
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const data_stage2 = sim.evaluate("transpose(concat( transpose( result_stage2[:,4] - phi0) * r0 / rad / km, ( transpose(result_stage2[:,1]) - r0) / km, 1 ))").toArray().map(function(e) { return {x: e[0], y: e[1]} })
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const data_unpowered = sim.evaluate("transpose(concat( transpose( result_unpowered[:,4] - phi0) * r0 / rad / km, ( transpose(result_unpowered[:,1]) - r0) / km, 1 ))").toArray().map(function(e) { return {x: e[0], y: e[1]} })
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window['chart'] = new Chart(document.getElementById('canvas1'), {
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type: 'line',
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data: {
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datasets: [{
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label: "Stage 1",
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data: data_stage1,
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fill: false,
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borderColor: "red",
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pointRadius: 0
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}, {
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label: "Interstage",
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data: data_interstage,
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fill: false,
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borderColor: "green",
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pointRadius: 0
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}, {
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label: "Stage 2",
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data: data_stage2,
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fill: false,
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borderColor: "orange",
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pointRadius: 0
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}, {
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label: "Unpowered",
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data: data_unpowered,
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fill: false,
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borderColor: "blue",
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pointRadius: 0
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}]
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},
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options: {
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scales: {
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xAxes: [{
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type: 'linear',
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position: 'bottom'
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}]
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}
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}
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})
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</script>
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</body>
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</html>
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