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Shepperd.cs
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Shepperd.cs
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using System;
using MechJebLib.Primitives;
namespace MechJebLib.Core.TwoBody
{
public static class Shepperd
{
// NOTE: this isn't a faithful reproduction of Shepperd's method, it works
// better than Shepperd's method and uses different constants in the continued
// faction, although it otherwise has almost the same "shape" of algorithm.
// I don't yet understand exactly why or how it works.
public static (V3 rf, V3 vf) Solve(double mu, double tau, V3 ri, V3 vi)
{
double tolerance = 1.0e-12;
double u = 0;
int imax = 50;
double umax = double.MaxValue;
double umin = double.MinValue;
double orbits = 0;
double tdesired = tau;
double threshold = tolerance * Math.Abs(tdesired);
double r0 = ri.magnitude;
double n0 = V3.Dot(ri, vi);
double beta = 2.0 * (mu / r0) - vi.sqrMagnitude;
if (beta != 0.0)
{
umax = 1.0 / Math.Sqrt(Math.Abs(beta));
umin = -1.0 / Math.Sqrt(Math.Abs(beta));
}
if (beta > 0.0)
{
orbits = beta * tau - 2 * n0;
orbits = 1 + orbits * Math.Sqrt(beta) / (Math.PI * mu);
orbits = Math.Floor(orbits / 2);
}
double uold = double.MinValue;
double dtold = double.MinValue;
double u0;
double u1 = 0.0;
double u2 = 0.0;
double u3;
double r1 = 0.0;
double q, n, r, l, s, d, gcf, k, gold, dt, slope, terror, ustep, h0, h1;
for (int i = 1; i < imax; i++)
{
q = beta * u * u;
q /= 1.0 + q;
n = 0;
r = 1;
l = 1;
s = 1;
d = 3;
gcf = 1;
k = -5;
gold = 0;
while (gcf != gold)
{
k = -k;
l += 2;
d += 4 * l;
n += (1 + k) * l;
r = d / (d - n * r * q);
s = (r - 1) * s;
gold = gcf;
gcf = gold + s;
}
h0 = 1 - 2 * q;
h1 = 2 * u * (1 - q);
u0 = 2 * h0 * h0 - 1;
u1 = 2 * h0 * h1;
u2 = 2 * h1 * h1;
u3 = 2 * h1 * u2 * gcf / 3;
if (orbits != 0)
{
u3 += 2 * Math.PI * orbits / (beta * Math.Sqrt(beta));
}
r1 = r0 * u0 + n0 * u1 + mu * u2;
dt = r0 * u1 + n0 * u2 + mu * u3;
slope = 4 * r1 / (1 + beta * u * u);
terror = tdesired - dt;
if (Math.Abs(terror) < threshold)
break;
if (i > 1 && u == uold)
break;
if (i > 1 && dt == dtold)
break;
uold = u;
dtold = dt;
ustep = terror / slope;
if (ustep > 0)
{
umin = u;
u += ustep;
if (u > umax)
{
u = (umin + umax) / 2;
}
}
else
{
umax = u;
u += ustep;
if (u < umin)
{
u = (umin + umax) / 2;
}
}
if (i == imax)
{
// FIXME: throw
}
}
double f = 1.0 - mu / r0 * u2;
double gg = 1.0 - mu / r1 * u2;
double g = r0 * u1 + n0 * u2;
double ff = -mu * u1 / (r0 * r1);
var rf = new V3();
var vf = new V3();
for (int i = 0; i < 3; i++)
{
rf[i] = f * ri[i] + g * vi[i];
vf[i] = ff * ri[i] + gg * vi[i];
}
return (rf, vf);
}
// The STM is a 6x6 matrix which we return decomposed into 4 3x3 matricies
//
// [ 𝛿r ] = [ stm00 stm01 ] [ r ]
// [ 𝛿v ] = [ stm10 stm11 ] [ v ]
//
// NOTE: this is a fairly faithful reproduction of Shepperd's method with a little added bisection method in
// case Newton's Method gets into trouble.
//
public static ( V3 rf, V3 vf, M3 stm00, M3 stm01, M3 stm10, M3 stm11) Solve2(double mu, double tau, V3 ri, V3 vi)
{
double tol = 1.0e-12;
double n0 = V3.Dot(ri, vi);
double r0 = ri.magnitude;
double beta = 2.0 * (mu / r0) - vi.sqrMagnitude;
double u = 0;
double umax = double.MaxValue;
double umin = double.MinValue;
M3 stm00 = M3.zero;
M3 stm01 = M3.zero;
M3 stm10 = M3.zero;
M3 stm11 = M3.zero;
if (beta != 0.0)
{
umax = 1.0 / Math.Sqrt(Math.Abs(beta));
}
else
{
umax = 1.0e24;
}
umin = -umax;
double delu = 0.0;
if (beta > 0.0)
{
double beta3 = beta * beta * beta;
double p = 2.0 * Math.PI * mu * 1 / Math.Sqrt(beta3);
double norb = Math.Truncate(1.0 / p * (tau + 0.5 * p - 2 * n0 / beta));
delu = 2.0 * norb * Math.PI * 1 / Math.Sqrt(beta3 * beta * beta);
}
double tsav = 1.0e99;
int niter = 0;
// kepler iteration loop
double q, u0, u1, u2, u3, uu, usav, dtdu, du, n, l, d, k, a, b, gcf, gsav, r1, t, terr;
while (true)
{
niter++;
q = beta * u * u;
q /= 1.0 + q;
u0 = 1.0 - 2 * q;
u1 = 2.0 * u * (1 - q);
// continued fraction iteration
n = 0;
l = 3;
d = 15;
k = -9;
a = 1;
b = 1;
gcf = 1;
while (true)
{
gsav = gcf;
k = -k;
l += 2;
d += 4 * l;
n += (1 + k) * l;
a = d / (d - n * a * q);
b = (a - 1) * b;
gcf += b;
if (Math.Abs(gcf - gsav) < tol)
break;
}
uu = 16.0 / 15.0 * u1 * u1 * u1 * u1 * u1 * gcf + delu;
u2 = 2.0 * u1 * u1;
u1 = 2.0 * u0 * u1;
u0 = 2.0 * u0 * u0 - 1.0;
u3 = beta * uu + u1 * u2 / 3.0;
r1 = r0 * u0 + n0 * u1 + mu * u2;
t = r0 * u1 + n0 * u2 + mu * u3;
dtdu = 4.0 * r1 * (1.0 - q);
// check for time convergence
if (Math.Abs(t - tsav) < tol)
break;
usav = u;
tsav = t;
terr = tau - t;
if (Math.Abs(terr) < Math.Abs(tau) * tol)
break;
du = terr / dtdu;
if (du < 0.0)
{
umax = u;
u += du;
if (u < umin)
u = 0.5 * (umin + umax);
}
else
{
umin = u;
u += du;
if (u > umax)
u = 0.5 * (umin + umax);
}
// check for independent variable convergence
if (Math.Abs(u - usav) < tol)
break;
// check for more than 20 iterations
if (niter > 20)
{
// FIXME: throw
}
}
double fm = -mu * u2 / r0;
double ggm = -mu * u2 / r1;
double f = 1.0 + fm;
double g = r0 * u1 + n0 * u2;
double ff = -mu * u1 / (r0 * r1);
double gg = 1.0 + ggm;
// compute final state vector
V3 rf = f * ri + g * vi;
V3 vf = ff * ri + gg * vi;
double w = g * u2 + 3 * mu * uu;
double a0 = mu / (r0 * r0 * r0);
double a1 = mu / (r1 * r1 * r1);
M3 m = M3.zero;
m[0, 0] = ff * (u0 / (r0 * r1) + 1.0 / (r0 * r0) + 1.0 / (r1 * r1));
m[0, 1] = (ff * u1 + ggm / r1) / r1;
m[0, 2] = ggm * u1 / r1;
m[1, 0] = -(ff * u1 + fm / r0) / r0;
m[1, 1] = -ff * u2;
m[1, 2] = -ggm * u2;
m[2, 0] = fm * u1 / r0;
m[2, 1] = fm * u2;
m[2, 2] = g * u2;
m[0, 0] -= a0 * a1 * w;
m[0, 2] -= a1 * w;
m[2, 0] -= a0 * w;
m[2, 2] -= w;
for (int i = 0; i < 3; i++)
{
double t001 = rf[i] * m[1, 0] + vf[i] * m[2, 0];
double t002 = rf[i] * m[1, 1] + vf[i] * m[2, 1];
double t011 = rf[i] * m[1, 1] + vf[i] * m[2, 1];
double t012 = rf[i] * m[1, 2] + vf[i] * m[2, 2];
double t101 = rf[i] * m[0, 0] + vf[i] * m[1, 0];
double t102 = rf[i] * m[0, 1] + vf[i] * m[1, 1];
double t111 = rf[i] * m[0, 1] + vf[i] * m[1, 1];
double t112 = rf[i] * m[0, 2] + vf[i] * m[1, 2];
for (int j = 0; j < 3; j++)
{
stm00[i, j] = t001 * ri[j] + t002 * vi[j];
stm01[i, j] = t011 * ri[j] + t012 * vi[j];
stm10[i, j] = -t101 * ri[j] - t102 * vi[j];
stm11[i, j] = -t111 * ri[j] - t112 * vi[j];
}
}
for (int i = 0; i < 3; i++)
{
stm00[i, i] += f;
stm01[i, i] += g;
stm10[i, i] += ff;
stm11[i, i] += gg;
}
return (rf, vf, stm00, stm01, stm10, stm11);
}
}
}