IodLambert.java

  1. /* Copyright 2002-2025 CS GROUP
  2.  * Licensed to CS GROUP (CS) under one or more
  3.  * contributor license agreements.  See the NOTICE file distributed with
  4.  * this work for additional information regarding copyright ownership.
  5.  * CS licenses this file to You under the Apache License, Version 2.0
  6.  * (the "License"); you may not use this file except in compliance with
  7.  * the License.  You may obtain a copy of the License at
  8.  *
  9.  *   http://www.apache.org/licenses/LICENSE-2.0
  10.  *
  11.  * Unless required by applicable law or agreed to in writing, software
  12.  * distributed under the License is distributed on an "AS IS" BASIS,
  13.  * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
  14.  * See the License for the specific language governing permissions and
  15.  * limitations under the License.
  16.  */
  17. package org.orekit.estimation.iod;

  19. import org.hipparchus.geometry.euclidean.threed.Vector3D;
  20. import org.hipparchus.util.FastMath;
  21. import org.orekit.errors.OrekitException;
  22. import org.orekit.errors.OrekitMessages;
  23. import org.orekit.estimation.measurements.PV;
  24. import org.orekit.estimation.measurements.Position;
  25. import org.orekit.frames.Frame;
  26. import org.orekit.orbits.CartesianOrbit;
  27. import org.orekit.orbits.Orbit;
  28. import org.orekit.time.AbsoluteDate;
  29. import org.orekit.utils.PVCoordinates;

  31. /**
  32.  * Lambert position-based Initial Orbit Determination (IOD) algorithm, assuming Keplerian motion.
  33.  * <p>
  34.  * An orbit is determined from two position vectors.
  35.  *
  36.  * References:
  37.  *  Battin, R.H., An Introduction to the Mathematics and Methods of Astrodynamics, AIAA Education, 1999.
  38.  *  Lancaster, E.R. and Blanchard, R.C., A Unified Form of Lambert’s Theorem, Goddard Space Flight Center, 1968.
  39.  * </p>
  40.  * @author Joris Olympio
  41.  * @since 8.0
  42.  */
  43. public class IodLambert {

  45.     /** gravitational constant. */
  46.     private final double mu;

  48.     /** Creator.
  49.      *
  50.      * @param mu gravitational constant
  51.      */
  52.     public IodLambert(final double mu) {
  53.         this.mu = mu;
  54.     }

  56.     /** Estimate an initial orbit from two position measurements.
  57.      * <p>
  58.      * The logic for setting {@code posigrade} and {@code nRev} is that the
  59.      * sweep angle Δυ travelled by the object between {@code t1} and {@code t2} is
  60.      * 2π {@code nRev +1} - α if {@code posigrade} is false and 2π {@code nRev} + α
  61.      * if {@code posigrade} is true, where α is the separation angle between
  62.      * {@code p1} and {@code p2}, which is always computed between 0 and π
  63.      * (because in 3D without a normal reference, vector angles cannot go past π).
  64.      * </p>
  65.      * <p>
  66.      * This implies that {@code posigrade} should be set to true if {@code p2} is
  67.      * located in the half orbit starting at {@code p1} and it should be set to
  68.      * false if {@code p2} is located in the half orbit ending at {@code p1},
  69.      * regardless of the number of periods between {@code t1} and {@code t2},
  70.      * and {@code nRev} should be set accordingly.
  71.      * </p>
  72.      * <p>
  73.      * As an example, if {@code t2} is less than half a period after {@code t1},
  74.      * then {@code posigrade} should be {@code true} and {@code nRev} should be 0.
  75.      * If {@code t2} is more than half a period after {@code t1} but less than
  76.      * one period after {@code t1}, {@code posigrade} should be {@code false} and
  77.      * {@code nRev} should be 0.
  78.      * </p>
  79.      * @param frame     measurements frame
  80.      * @param posigrade flag indicating the direction of motion
  81.      * @param nRev      number of revolutions
  82.      * @param p1        first position measurement
  83.      * @param p2        second position measurement
  84.      * @return an initial Keplerian orbit estimation at the first observation date t1
  85.      * @since 11.0
  86.      */
  87.     public Orbit estimate(final Frame frame, final boolean posigrade,
  88.                           final int nRev, final Position p1,  final Position p2) {
  89.         return estimate(frame, posigrade, nRev,
  90.                         p1.getPosition(), p1.getDate(), p2.getPosition(), p2.getDate());
  91.     }

  93.     /** Estimate an initial orbit from two PV measurements.
  94.      * <p>
  95.      * The logic for setting {@code posigrade} and {@code nRev} is that the
  96.      * sweep angle Δυ travelled by the object between {@code t1} and {@code t2} is
  97.      * 2π {@code nRev +1} - α if {@code posigrade} is false and 2π {@code nRev} + α
  98.      * if {@code posigrade} is true, where α is the separation angle between
  99.      * {@code p1} and {@code p2}, which is always computed between 0 and π
  100.      * (because in 3D without a normal reference, vector angles cannot go past π).
  101.      * </p>
  102.      * <p>
  103.      * This implies that {@code posigrade} should be set to true if {@code p2} is
  104.      * located in the half orbit starting at {@code p1} and it should be set to
  105.      * false if {@code p2} is located in the half orbit ending at {@code p1},
  106.      * regardless of the number of periods between {@code t1} and {@code t2},
  107.      * and {@code nRev} should be set accordingly.
  108.      * </p>
  109.      * <p>
  110.      * As an example, if {@code t2} is less than half a period after {@code t1},
  111.      * then {@code posigrade} should be {@code true} and {@code nRev} should be 0.
  112.      * If {@code t2} is more than half a period after {@code t1} but less than
  113.      * one period after {@code t1}, {@code posigrade} should be {@code false} and
  114.      * {@code nRev} should be 0.
  115.      * </p>
  116.      * @param frame     measurements frame
  117.      * @param posigrade flag indicating the direction of motion
  118.      * @param nRev      number of revolutions
  119.      * @param pv1       first PV measurement
  120.      * @param pv2       second PV measurement
  121.      * @return an initial Keplerian orbit estimation at the first observation date t1
  122.      * @since 12.0
  123.      */
  124.     public Orbit estimate(final Frame frame, final boolean posigrade,
  125.                           final int nRev, final PV pv1,  final PV pv2) {
  126.         return estimate(frame, posigrade, nRev,
  127.                         pv1.getPosition(), pv1.getDate(), pv2.getPosition(), pv2.getDate());
  128.     }

  130.     /** Estimate a Keplerian orbit given two position vectors and a duration.
  131.      * <p>
  132.      * The logic for setting {@code posigrade} and {@code nRev} is that the
  133.      * sweep angle Δυ travelled by the object between {@code t1} and {@code t2} is
  134.      * 2π {@code nRev +1} - α if {@code posigrade} is false and 2π {@code nRev} + α
  135.      * if {@code posigrade} is true, where α is the separation angle between
  136.      * {@code p1} and {@code p2}, which is always computed between 0 and π
  137.      * (because in 3D without a normal reference, vector angles cannot go past π).
  138.      * </p>
  139.      * <p>
  140.      * This implies that {@code posigrade} should be set to true if {@code p2} is
  141.      * located in the half orbit starting at {@code p1} and it should be set to
  142.      * false if {@code p2} is located in the half orbit ending at {@code p1},
  143.      * regardless of the number of periods between {@code t1} and {@code t2},
  144.      * and {@code nRev} should be set accordingly.
  145.      * </p>
  146.      * <p>
  147.      * As an example, if {@code t2} is less than half a period after {@code t1},
  148.      * then {@code posigrade} should be {@code true} and {@code nRev} should be 0.
  149.      * If {@code t2} is more than half a period after {@code t1} but less than
  150.      * one period after {@code t1}, {@code posigrade} should be {@code false} and
  151.      * {@code nRev} should be 0.
  152.      * </p>
  153.      * @param frame     frame
  154.      * @param posigrade flag indicating the direction of motion
  155.      * @param nRev      number of revolutions
  156.      * @param p1        position vector 1
  157.      * @param t1        date of observation 1
  158.      * @param p2        position vector 2
  159.      * @param t2        date of observation 2
  160.      * @return  an initial Keplerian orbit estimate at the first observation date t1
  161.      */
  162.     public Orbit estimate(final Frame frame, final boolean posigrade,
  163.                           final int nRev,
  164.                           final Vector3D p1, final AbsoluteDate t1,
  165.                           final Vector3D p2, final AbsoluteDate t2) {

  167.         final double r1 = p1.getNorm();
  168.         final double r2 = p2.getNorm();
  169.         final double tau = t2.durationFrom(t1); // in seconds

  171.         // Exception if t2 < t1
  172.         if (tau < 0.0) {
  173.             throw new OrekitException(OrekitMessages.NON_CHRONOLOGICAL_DATES_FOR_OBSERVATIONS, t1, t2, -tau);
  174.         }

  176.         // normalizing constants
  177.         final double R = FastMath.max(r1, r2); // in m
  178.         final double V = FastMath.sqrt(mu / R);  // in m/s
  179.         final double T = R / V; // in seconds

  181.         // sweep angle
  182.         double dth = Vector3D.angle(p1, p2);
  183.         // compute the number of revolutions
  184.         if (!posigrade) {
  185.             dth = 2 * FastMath.PI - dth;
  186.         }

  188.         // velocity vectors in the orbital plane, in the R-T frame
  189.         final double[] Vdep = new double[2];

  191.         // call Lambert's problem solver
  192.         final boolean exitflag = solveLambertPb(r1 / R, r2 / R, dth, tau / T, nRev, Vdep);

  194.         if (exitflag) {
  195.             // basis vectors
  196.             // normal to the orbital arc plane
  197.             final Vector3D Pn = p1.crossProduct(p2);
  198.             // perpendicular to the radius vector, in the orbital arc plane
  199.             final Vector3D Pt = Pn.crossProduct(p1);

  201.             // tangential velocity norm
  202.             double RT = Pt.getNorm();
  203.             if (!posigrade) {
  204.                 RT = -RT;
  205.             }

  207.             // velocity vector at P1
  208.             final Vector3D Vel1 = new Vector3D(V * Vdep[0] / r1, p1,
  209.                                                V * Vdep[1] / RT, Pt);

  211.             // compute the equivalent Cartesian orbit
  212.             return new CartesianOrbit(new PVCoordinates(p1, Vel1), frame, t1, mu);
  213.         }

  215.         return null;
  216.     }

  218.     /** Lambert's solver.
  219.      * Assume mu=1.
  220.      *
  221.      * @param r1 radius 1
  222.      * @param r2  radius 2
  223.      * @param dth sweep angle
  224.      * @param tau time of flight
  225.      * @param mRev number of revs
  226.      * @param V1 velocity at departure in (T, N) basis
  227.      * @return something
  228.      */
  229.     boolean solveLambertPb(final double r1, final double r2, final double dth, final double tau,
  230.                            final int mRev, final double[] V1) {
  231.         // decide whether to use the left or right branch (for multi-revolution
  232.         // problems), and the long- or short way.
  233.         final boolean leftbranch = dth < FastMath.PI;
  234.         int longway = 1;
  235.         if (dth > FastMath.PI) {
  236.             longway = -1;
  237.         }

  239.         final int m = FastMath.abs(mRev);
  240.         final double rtof = FastMath.abs(tau);

  242.         // non-dimensional chord ||r2-r1||
  243.         final double chord = FastMath.sqrt(r1 * r1 + r2 * r2 - 2 * r1 * r2 * FastMath.cos(dth));

  245.         // non-dimensional semi-perimeter of the triangle
  246.         final double speri = 0.5 * (r1 + r2 + chord);

  248.         // minimum energy ellipse semi-major axis
  249.         final double minSma = speri / 2.;

  251.         // lambda parameter (Eq 7.6)
  252.         final double lambda = longway * FastMath.sqrt(1 - chord / speri);

  254.         // reference tof value for the Newton solver
  255.         final double logt = FastMath.log(rtof);

  257.         // initialisation of the iterative root finding process (secant method)
  258.         // initial values
  259.         //  -1 < x < 1  =>  Elliptical orbits
  260.         //  x = 1           Parabolic orbit
  261.         //  x > 1           Hyperbolic orbits
  262.         final double in1;
  263.         final double in2;
  264.         double x1;
  265.         double x2;
  266.         if (m == 0) {
  267.             // one revolution, one solution. Define the left and right asymptotes.
  268.             in1 = -0.6523333;
  269.             in2 = 0.6523333;
  270.             x1   = FastMath.log(1 + in1);
  271.             x2   = FastMath.log(1 + in2);
  272.         } else {
  273.             // select initial values, depending on the branch
  274.             if (!leftbranch) {
  275.                 in1 = -0.523334;
  276.                 in2 = -0.223334;
  277.             } else {
  278.                 in1 = 0.723334;
  279.                 in2 = 0.523334;
  280.             }
  281.             x1 = FastMath.tan(in1 * 0.5 * FastMath.PI);
  282.             x2 = FastMath.tan(in2 * 0.5 * FastMath.PI);
  283.         }

  285.         // initial estimates for the tof
  286.         final double tof1 = timeOfFlight(in1, longway, m, minSma, speri, chord);
  287.         final double tof2 = timeOfFlight(in2, longway, m, minSma, speri, chord);

  289.         // initial bounds for y
  290.         double y1;
  291.         double y2;
  292.         if (m == 0) {
  293.             y1 = FastMath.log(tof1) - logt;
  294.             y2 = FastMath.log(tof2) - logt;
  295.         } else {
  296.             y1 = tof1 - rtof;
  297.             y2 = tof2 - rtof;
  298.         }

  300.         // Solve for x with the secant method
  301.         double err = 1e20;
  302.         int iterations = 0;
  303.         final double tol = 1e-13;
  304.         final int maxiter = 50;
  305.         double xnew = 0;
  306.         while (err > tol && iterations < maxiter) {
  307.             // new x
  308.             xnew = (x1 * y2 - y1 * x2) / (y2 - y1);

  310.             // evaluate new time of flight
  311.             final double x;
  312.             if (m == 0) {
  313.                 x = FastMath.exp(xnew) - 1;
  314.             } else {
  315.                 x = FastMath.atan(xnew) * 2 / FastMath.PI;
  316.             }

  318.             final double tof = timeOfFlight(x, longway, m, minSma, speri, chord);

  320.             // new value of y
  321.             final double ynew;
  322.             if (m == 0) {
  323.                 ynew = FastMath.log(tof) - logt;
  324.             } else {
  325.                 ynew = tof - rtof;
  326.             }

  328.             // save previous and current values for the next iteration
  329.             x1 = x2;
  330.             x2 = xnew;
  331.             y1 = y2;
  332.             y2 = ynew;

  334.             // update error
  335.             err = FastMath.abs(x1 - xnew);

  337.             // increment number of iterations
  338.             ++iterations;
  339.         }

  341.         // failure to converge
  342.         if (err > tol) {
  343.             return false;
  344.         }

  346.         // convert converged value of x
  347.         final double x;
  348.         if (m == 0) {
  349.             x = FastMath.exp(xnew) - 1;
  350.         } else {
  351.             x = FastMath.atan(xnew) * 2 / FastMath.PI;
  352.         }

  354.         // Solution for the semi-major axis (Eq. 7.20)
  355.         final double sma = minSma / (1 - x * x);

  357.         // compute velocities
  358.         final double eta;
  359.         if (x < 1) {
  360.             // ellipse, Eqs. 7.7, 7.17
  361.             final double alfa = 2 * FastMath.acos(x);
  362.             final double beta = longway * 2 * FastMath.asin(FastMath.sqrt((speri - chord) / (2. * sma)));
  363.             final double psi  = (alfa - beta) / 2;
  364.             // Eq. 7.21
  365.             final double sinPsi = FastMath.sin(psi);
  366.             final double etaSq = 2 * sma * sinPsi * sinPsi / speri;
  367.             eta  = FastMath.sqrt(etaSq);
  368.         } else {
  369.             // hyperbola
  370.             final double gamma = 2 * FastMath.acosh(x);
  371.             final double delta = longway * 2 * FastMath.asinh(FastMath.sqrt((chord - speri) / (2 * sma)));
  372.             //
  373.             final double psi  = (gamma - delta) / 2.;
  374.             final double sinhPsi = FastMath.sinh(psi);
  375.             final double etaSq = -2 * sma * sinhPsi * sinhPsi / speri;
  376.             eta  = FastMath.sqrt(etaSq);
  377.         }

  379.         // radial and tangential directions for departure velocity (Eq. 7.36)
  380.         final double VR1 = (1. / eta) * FastMath.sqrt(1. / minSma) * (2 * lambda * minSma / r1 - (lambda + x * eta));
  381.         final double VT1 = (1. / eta) * FastMath.sqrt(1. / minSma) * FastMath.sqrt(r2 / r1) * FastMath.sin(dth / 2);
  382.         V1[0] = VR1;
  383.         V1[1] = VT1;

  385.         return true;
  386.     }

  388.     /** Compute the time of flight of a given arc of orbit.
  389.      * The time of flight is evaluated via the Lagrange expression.
  390.      *
  391.      * @param x          x
  392.      * @param longway    solution number; the long way or the short war
  393.      * @param mrev       number of revolutions of the arc of orbit
  394.      * @param minSma     minimum possible semi-major axis
  395.      * @param speri      semi-parameter of the arc of orbit
  396.      * @param chord      chord of the arc of orbit
  397.      * @return the time of flight for the given arc of orbit
  398.      */
  399.     private double timeOfFlight(final double x, final int longway, final int mrev, final double minSma,
  400.                                 final double speri, final double chord) {

  402.         final double a = minSma / (1 - x * x);

  404.         final double tof;
  405.         if (FastMath.abs(x) < 1) {
  406.             // Lagrange form of the time of flight equation Eq. (7.9)
  407.             // elliptical orbit (note: mu = 1)
  408.             final double beta = longway * 2 * FastMath.asin(FastMath.sqrt((speri - chord) / (2. * a)));
  409.             final double alfa = 2 * FastMath.acos(x);
  410.             tof = a * FastMath.sqrt(a) * ((alfa - FastMath.sin(alfa)) - (beta - FastMath.sin(beta)) + 2 * FastMath.PI * mrev);
  411.         } else {
  412.             // hyperbolic orbit
  413.             final double alfa = 2 * FastMath.acosh(x);
  414.             final double beta = longway * 2 * FastMath.asinh(FastMath.sqrt((speri - chord) / (-2. * a)));
  415.             tof = -a * FastMath.sqrt(-a) * ((FastMath.sinh(alfa) - alfa) - (FastMath.sinh(beta) - beta));
  416.         }

  418.         return tof;
  419.     }
  420. }
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