/
Gooding.cs
553 lines (485 loc) · 18.3 KB
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Gooding.cs
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/*
* Copyright Lamont Granquist, Sebastien Gaggini and the MechJeb contributors
* SPDX-License-Identifier: LicenseRef-PD-hp OR Unlicense OR CC0-1.0 OR 0BSD OR MIT-0 OR MIT OR LGPL-2.1+
*/
using System;
using System.Runtime.CompilerServices;
using MechJebLib.Primitives;
using MechJebLib.Utils;
using static MechJebLib.Statics;
using static System.Math;
// ReSharper disable InconsistentNaming
// ReSharper disable CompareOfFloatsByEqualityOperator
#nullable enable
namespace MechJebLib.Core.Lambert
{
public static class Gooding
{
/*
* mu = gravitational parameter of central body
* r1 = position at t0
* v1 = velocity at t0
* r2 = position at t1
* tof = time of flight (t1 - t0) (+ posigrade "shortway", - retrograde "longway")
* nrev = number of full revolutions (+ left-branch, - right-branch for nrev != 0)
* Vi = initial velocity vector of transfer orbit
* Vf = final velocity vector of transfer orbit
*/
public static (V3 Vi, V3 Vf) Solve(double mu, V3 r1, V3 v1, V3 r2, double tof, int nrev)
{
/* most of this function lifted from https://www.mathworks.com/matlabcentral/fileexchange/39530-lambert-s-problem/content/glambert.m */
// if we don't catch this edge condition, the solver will spin forever (internal state will NaN and produce great sadness)
if (tof == 0)
throw new Exception("MechJeb's Gooding Lambert Solver does not support zero time of flight (teleportation)");
V3 Vi, Vf;
V3 ur1xv1 = V3.Cross(r1, v1).normalized;
V3 ux1 = r1.normalized;
V3 ux2 = r2.normalized;
V3 uz1 = V3.Cross(ux1, ux2).normalized;
/* calculate the minimum transfer angle (radians) */
double theta = SafeAcos(V3.Dot(ux1, ux2));
/* calculate the angle between the orbit normal of the initial orbit and the fundamental reference plane */
double angle_to_on = SafeAcos(V3.Dot(ur1xv1, uz1));
/* if angle to orbit normal is greater than 90 degrees and posigrade orbit, then flip the orbit normal and the transfer angle */
if (angle_to_on > 0.5 * PI && tof > 0.0)
{
theta = TAU - theta;
uz1 = -uz1;
}
if (angle_to_on < 0.5 * PI && tof < 0.0)
{
theta = TAU - theta;
uz1 = -uz1;
}
V3 uz2 = uz1;
V3 uy1 = V3.Cross(uz1, ux1).normalized;
V3 uy2 = V3.Cross(uz2, ux2).normalized;
theta += TAU * Abs(nrev);
double VR11, VT11, VR12, VT12;
double VR21, VT21, VR22, VT22;
int n;
(n, VR11, VT11, VR12, VT12, VR21, VT21, VR22, VT22) = VLAMB(mu, r1.magnitude, r2.magnitude, theta, tof);
if (nrev > 0)
{
switch (n)
{
case -1:
throw new Exception("Gooding Solver found no tminimum");
case 0:
throw new Exception("Gooding Solver found no solution time");
}
}
/* compute transfer orbit initial and final velocity vectors */
if (nrev > 0 && n > 1)
{
Vi = VR21 * ux1 + VT21 * uy1;
Vf = VR22 * ux2 + VT22 * uy2;
}
else
{
Vi = VR11 * ux1 + VT11 * uy1;
Vf = VR12 * ux2 + VT12 * uy2;
}
return (Vi, Vf);
}
/*
* Goodings Method
*
* MMMMMmmmmmm..... Smells like Fortran....
*
* Shield your eyes lest ye be blinded by goto statements...
*
* Keep in mind that Gooding optimized the math to reduce loss of precision so "cleaning up" these functions without knowing the values
* that the variables typically hold could result in two very small or very large numbers being multiplied together and resultant loss
* of precision, and that rearrangement will make it incredibly difficult to spot simple transcription typos. It has been deliberately
* kept as super-fugly looking C# code for those reasons.
*/
[MethodImpl(MethodImplOptions.AggressiveInlining)]
internal static ( int N, double VR11, double VT11, double VR12, double VT12, double VR21, double VT21, double VR22,
double VT22) VLAMB(double GM, double R1, double R2, double TH, double TDELT)
{
Check.Finite(GM);
Check.Finite(R1);
Check.Finite(R1);
Check.Finite(TH);
Check.Finite(TDELT);
double VR11 = 0.0, VT11 = 0.0, VR12 = 0.0, VT12 = 0.0;
double VR21 = 0.0, VT21 = 0.0, VR22 = 0.0, VT22 = 0.0;
int M = Convert.ToInt32(Floor(TH / (2.0 * PI)));
double THR2 = TH / 2.0 - M * PI;
double DR = R1 - R2;
double R1R2 = R1 * R2;
double R1R2TH = 4.0 * R1R2 * Pow(Sin(THR2), 2);
double CSQ = Pow(DR, 2) + R1R2TH;
double C = Sqrt(CSQ);
double S = (R1 + R2 + C) / 2.0;
double GMS = Sqrt(GM * S / 2.0);
double QSQFM1 = C / S;
double Q = Sqrt(R1R2) * Cos(THR2) / S;
double RHO;
double SIG;
if (C != 0.0)
{
RHO = DR / C;
SIG = R1R2TH / CSQ;
}
else
{
RHO = 0.0;
SIG = 1.0;
}
double T = 4.0 * GMS * TDELT / Pow(S, 2);
(int N, double X1, double X2) = XLAMB(M, Q, QSQFM1, T);
for (int I = 1; I <= N; I++)
{
double X = I == 1 ? X1 : X2;
double QZMINX;
double QZPLX;
double ZPLQX;
(_, QZMINX, QZPLX, ZPLQX) = TLAMB(M, Q, QSQFM1, X, -1);
double VT2 = GMS * ZPLQX * Sqrt(SIG);
double VR1 = GMS * (QZMINX - QZPLX * RHO) / R1;
double VT1 = VT2 / R1;
double VR2 = -GMS * (QZMINX + QZPLX * RHO) / R2;
VT2 /= R2;
if (I == 1)
{
VR11 = VR1;
VT11 = VT1;
VR12 = VR2;
VT12 = VT2;
}
else
{
VR21 = VR1;
VT21 = VT1;
VR22 = VR2;
VT22 = VT2;
}
}
return (N, VR11, VT11, VR12, VT12, VR21, VT21, VR22, VT22);
}
[MethodImpl(MethodImplOptions.AggressiveInlining)]
private static ( int N, double X, double XPL) XLAMB(int M, double Q, double QSQFM1, double TIN)
{
Check.Finite(M);
Check.Finite(Q);
Check.Finite(QSQFM1);
Check.Finite(TIN);
const double TOL = 3e-7;
const double C0 = 1.7;
const double C1 = 0.5;
const double C2 = 0.03;
const double C3 = 0.15;
const double C41 = 1.0;
const double C42 = 0.24;
double THR2 = Atan2(QSQFM1, 2.0 * Q) / PI;
double T, T0, DT, D2T, D3T;
double D2T2 = 0.0;
double TMIN = 0.0;
double TDIFF;
double TDIFFM = 0.0;
double XM = 0.0;
double W;
double X = 0.0;
double XPL = 0.0;
int N;
if (M == 0)
{
/* "SINGLE-REV STARTER FROM T (AT X = 0) & BILINEAR (USUALLY)" -- Gooding */
N = 1;
(T0, _, _, _) = TLAMB(M, Q, QSQFM1, 0.0, 0);
TDIFF = TIN - T0;
if (TDIFF <= 0.0)
{
X = T0 * TDIFF / (-4.0 * TIN);
/* "-4 IS THE VALUE OF DT, FOR X = 0" -- Gooding */
}
else
{
X = -TDIFF / (TDIFF + 4.0);
W = X + C0 * Sqrt(2.0 * (1.0 - THR2));
if (W < 0.0)
X -= Sqrt(Pow(-W, 1.0 / 8.0)) * (X + Sqrt(TDIFF / (TDIFF + 1.5 * T0)));
W = 4.0 / (4.0 + TDIFF);
X *= 1.0 + X * (C1 * W - C2 * X * Sqrt(W));
}
}
else
{
/* "WITH MUTIREVS, FIRST GET T(MIN) AS BASIS FOR STARTER */
XM = 1.0 / (1.5 * (M + 0.5) * PI);
if (THR2 < 0.5)
XM = Pow(2.0 * THR2, 1.0 / 8.0) * XM;
if (THR2 > 0.5)
XM = (2.0 - Pow(2.0 - 2.0 * THR2, 1.0 / 8.0)) * XM;
/* "STARTER FOR TMIN" */
for (int I = 1; I <= 12; I++)
{
(TMIN, DT, D2T, D3T) = TLAMB(M, Q, QSQFM1, XM, 3);
if (D2T == 0.0)
goto Two;
double XMOLD = XM;
XM -= DT * D2T / (D2T * D2T - DT * D3T / 2.0);
double XTEST = Abs(XMOLD / XM - 1.0);
if (XTEST <= TOL)
goto Two;
}
N = -1;
return (N, X, XPL);
/* "(BREAK OFF & EXIT IF TMIN NOT LOCATED - SHOULD NEVER HAPPEN)" */
/* "NOW PROCEED FROM T(MIN) TO FULL STARTER" -- Gooding */
Two:
TDIFFM = TIN - TMIN;
if (TDIFFM < 0.0)
{
N = 0;
return (N, X, XPL);
/* "EXIT IF NO SOLUTION ALTREADY FROM X(TMIN)" -- Gooding */
}
if (TDIFFM == 0.0)
{
X = XM;
N = 1;
return (N, X, XPL);
/* "EXIT IF UNIQUE SOLUTION ALREADY FROM X(TMIN) -- Gooding */
}
N = 3;
if (D2T == 0.0)
D2T = 6.0 * M * PI;
X = Sqrt(TDIFFM / (D2T / 2.0 + TDIFFM / Pow(1.0 - XM, 2.0)));
W = XM + X;
W = W * 4.0 / (4.0 + TDIFFM) + Pow(1.0 - W, 2.0);
X = X * (1.0 - (1.0 + M + C41 * (THR2 - 0.5)) / (1.0 + C3 * M) * X * (C1 * W + C2 * X * Sqrt(W))) + XM;
D2T2 = D2T / 2.0;
if (X >= 1.0)
{
N = 1;
goto Three;
}
/* "(NO FINITE SOLUTION WITH X > XM)" -- Gooding */
}
/* "(NOW HAVE A STARTER, SO PROCEED BY HALLEY)" -- Gooding */
Five:
for (int I = 1; I <= 3; I++)
{
(T, DT, D2T, _) = TLAMB(M, Q, QSQFM1, X, 2);
T = TIN - T;
if (DT != 0.0)
X += T * DT / (DT * DT + T * D2T / 2.0);
}
if (N != 3)
return (N, X, XPL);
/* "(EXIT IF ONLY ONE SOLUTION, NORMALLY WHEN M = 0)" */
N = 2;
XPL = X;
Three:
/* "(SECOND MULTI-REV STARTER)" */
(T0, _, _, _) = TLAMB(M, Q, QSQFM1, 0.0, 0);
double TDIFF0 = T0 - TMIN;
TDIFF = TIN - T0;
if (TDIFF <= 0.0)
{
X = XM - Sqrt(TDIFFM / (D2T2 - TDIFFM * (D2T2 / TDIFF0 - 1.0 / Pow(XM, 2))));
}
else
{
X = -TDIFF / (TDIFF + 4.0);
W = X + C0 * Sqrt(2.0 * (1.0 - THR2));
if (W < 0.0)
X -= Sqrt(Pow(-W, 1.0 / 8.0)) * (X + Sqrt(TDIFF / (TDIFF + 1.5 * T0)));
W = 4.0 / (4.0 + TDIFF);
X *= 1.0 + (1.0 + M + C42 * (THR2 - 0.5)) / (1.0 + C3 * M) * X * (C1 * W - C2 * X * Sqrt(W));
if (X <= -1.0)
{
N -= 1;
/* "(NO FINITE SOLUTION WITH X < XM)" -- Gooding */
if (N == 1)
X = XPL;
}
}
goto Five;
}
[MethodImpl(MethodImplOptions.AggressiveInlining)]
private static ( double T, double DT, double D2T, double D3T) TLAMB(int M, double Q, double QSQFM1, double X, int N)
{
Check.Finite(X);
Check.Finite(QSQFM1);
Check.Finite(Q);
Check.Finite(M);
Check.Finite(N);
const double SW = 0.4;
bool LM1 = N == -1;
bool L1 = N >= 1;
bool L2 = N >= 2;
bool L3 = N == 3;
double QSQ = Q * Q;
double XSQ = X * X;
double U = (1.0 - X) * (1.0 + X);
double T = 0.0;
double DT = 0.0;
double D2T = 0.0;
double D3T = 0.0;
if (!LM1)
{
/* "NEEDED IF SERIES AND OTHERWISE USEFUL WHEN Z = 0" -- Gooding */
DT = 0.0;
D2T = 0.0;
D3T = 0.0;
}
if (LM1 || M > 0.0 || X < 0.0 || Abs(U) > SW)
{
/* "DIRECT COMPUTATION (NOT SERIES)" -- Gooding */
double Y = Sqrt(Abs(U));
double Z = Sqrt(QSQFM1 + QSQ * XSQ);
double QX = Q * X;
double A = 0.0;
double B = 0.0;
double AA = 0.0;
double BB = 0.0;
if (QX <= 0.0)
{
A = Z - QX;
B = Q * Z - X;
}
if (QX < 0.0 && LM1)
{
AA = QSQFM1 / A;
BB = QSQFM1 * (QSQ * U - XSQ) / B;
}
if ((QX == 0.0 && LM1) || QX > 0.0)
{
AA = Z + QX;
BB = Q * Z + X;
}
if (QX > 0.0)
{
A = QSQFM1 / AA;
B = QSQFM1 * (QSQ * U - XSQ) / BB;
}
if (!LM1)
{
double G;
if (QX * U >= 0.0)
{
G = X * Z + Q * U;
}
else
{
G = (XSQ - QSQ * U) / (X * Z - Q * U);
}
double F = A * Y;
if (X <= 1.0)
{
T = M * PI + Atan2(F, G);
}
else
{
if (F > SW)
{
T = Log(F + G);
}
else
{
double FG1 = F / (G + 1.0);
double TERM = 2.0 * FG1;
double FG1SQ = FG1 * FG1;
T = TERM;
double TWOI1 = 1.0;
double TOLD;
do
{
TWOI1 += 2.0;
TERM *= FG1SQ;
TOLD = T;
T += TERM / TWOI1;
} while (T != TOLD); /* "CONTINUE LOOPING FOR THE INVERSE TANH" -- Gooding */
}
}
T = 2.0 * (T / Y + B) / U;
if (L1 && Z != 0.0)
{
double QZ = Q / Z;
double QZ2 = QZ * QZ;
QZ *= QZ2;
DT = (3.0 * X * T - 4.0 * (A + QX * QSQFM1) / Z) / U;
if (L2)
{
D2T = (3.0 * T + 5.0 * X * DT + 4.0 * QZ * QSQFM1) / U;
}
if (L3)
{
D3T = (8.0 * DT + 7.0 * X * D2T - 12.0 * QZ * QZ2 * X * QSQFM1) / U;
}
}
}
else
{
DT = B;
D2T = BB;
D3T = AA;
}
}
else
{
/* "COMPUTE BY SERIES" -- Gooding */
double U0I = 1.0;
double U1I = 0.0;
double U2I = 0.0;
double U3I = 0.0;
if (L1)
U1I = 1.0;
if (L2)
U2I = 1.0;
if (L3)
U3I = 1.0;
double TERM = 4.0;
double TQ = Q * QSQFM1;
int I = 0;
double TQSUM = 0.0;
if (Q < 0.5)
TQSUM = 1.0 - Q * QSQ;
if (Q >= 0.5)
TQSUM = (1.0 / (1.0 + Q) + Q) * QSQFM1;
double TTMOLD = TERM / 3.0;
T = TTMOLD * TQSUM;
double TOLD;
do
{
I++;
int P = I;
U0I *= U;
if (L1 && I > 1)
U1I *= U;
if (L2 && I > 2)
U2I *= U;
if (L3 && I > 3)
U3I *= U;
TERM = TERM * (P - 0.5) / P;
TQ *= QSQ;
TQSUM += TQ;
TOLD = T;
double TTERM = TERM / (2.0 * P + 3.0);
double TQTERM = TTERM * TQSUM;
T -= U0I * ((1.5 * P + 0.25) * TQTERM / (P * P - 0.25) - TTMOLD * TQ);
TTMOLD = TTERM;
TQTERM *= P;
if (L1)
DT += TQTERM * U1I;
if (L2)
D2T += TQTERM * U2I * (P - 1.0);
if (L3)
D3T += TQTERM * U3I * (P - 1.0) * (P - 2.0);
} while (I < N || T != TOLD);
if (L3)
D3T = 8.0 * X * (1.5 * D2T - XSQ * D3T);
if (L2)
D2T = 2.0 * (2.0 * XSQ * D2T - DT);
if (L1)
DT = -2.0 * X * DT;
T /= XSQ;
}
return (T, DT, D2T, D3T);
}
}
}