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    * contributor license agreements.  See the NOTICE file distributed with
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    * CS licenses this file to You under the Apache License, Version 2.0
    * (the "License"); you may not use this file except in compliance with
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    *   http://www.apache.org/licenses/LICENSE-2.0
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   * Unless required by applicable law or agreed to in writing, software
   * distributed under the License is distributed on an "AS IS" BASIS,
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  package org.orekit.bodies;
  
  import org.hipparchus.analysis.UnivariateFunction;
  import org.hipparchus.analysis.solvers.AllowedSolution;
  import org.hipparchus.analysis.solvers.BracketingNthOrderBrentSolver;
  import org.hipparchus.analysis.solvers.UnivariateSolverUtils;
  import org.hipparchus.geometry.euclidean.threed.Vector3D;
  import org.hipparchus.util.FastMath;
  import org.orekit.frames.CR3BPRotatingFrame;
  import org.orekit.frames.Frame;
  import org.orekit.frames.Transform;
  import org.orekit.time.AbsoluteDate;
  import org.orekit.utils.AbsolutePVCoordinates;
  import org.orekit.utils.LagrangianPoints;
  import org.orekit.utils.PVCoordinates;
  
  /**
   * Class creating, from two different celestial bodies, the corresponding system
   * with respect to the Circular Restricted Three Body problem hypotheses.
   * @see "Dynamical systems, the three-body problem, and space mission design, Koon, Lo, Marsden, Ross"
   * @author Vincent Mouraux
   * @author William Desprats
   * @since 10.2
   */
  public class CR3BPSystem {
  
      /** Relative accuracy on position for solver. */
      private static final double RELATIVE_ACCURACY = 1e-14;
  
      /** Absolute accuracy on position for solver (1mm). */
      private static final double ABSOLUTE_ACCURACY = 1e-3;
  
      /** Function value accuracy for solver (set to 0 so we rely only on position for convergence). */
      private static final double FUNCTION_ACCURACY = 0;
  
      /** Maximal order for solver. */
      private static final int MAX_ORDER = 5;
  
      /** Maximal number of evaluations for solver. */
      private static final int MAX_EVALUATIONS = 10000;
  
      /** Mass ratio. */
      private final double mu;
  
      /** Distance between the two primaries, meters. */
      private final double dDim;
  
      /** Orbital Velocity of m1, m/s. */
      private final double vDim;
  
      /** Orbital Period of m1 and m2, seconds. */
      private final double tDim;
  
      /** CR3BP System name. */
      private final String name;
  
      /** Rotating Frame for the system. */
      private final Frame rotatingFrame;
  
      /** Primary body. */
      private final CelestialBody primaryBody;
  
      /** Secondary body. */
      private final CelestialBody secondaryBody;
  
      /** L1 Point position in the rotating frame. */
      private Vector3D l1Position;
  
      /** L2 Point position in the rotating frame. */
      private Vector3D l2Position;
  
      /** L3 Point position in the rotating frame. */
      private Vector3D l3Position;
  
      /** L4 Point position in the rotating frame. */
      private Vector3D l4Position;
  
      /** L5 Point position in the rotating frame. */
      private Vector3D l5Position;
  
      /** Distance between a L1 and the secondaryBody. */
      private final double gamma1;
  
     /** Distance between a L2 and the secondaryBody. */
     private final double gamma2;
 
     /** Distance between a L3 and the primaryBody. */
     private final double gamma3;
 
     /**
      * Simple constructor.
      * <p>
      * Standard constructor to use to define a CR3BP System. Mass ratio is
      * calculated from JPL data.
      * </p>
      * @param primaryBody Primary Body in the CR3BP System
      * @param secondaryBody Secondary Body in the CR3BP System
      * @param a Semi-Major Axis of the secondary body
      */
     public CR3BPSystem(final CelestialBody primaryBody, final CelestialBody secondaryBody, final double a) {
         this(primaryBody, secondaryBody, a, secondaryBody.getGM() / (secondaryBody.getGM() + primaryBody.getGM()));
     }
 
     /**
      * Simple constructor.
      * <p>
      * This constructor can be used to define a CR3BP System if the user wants
      * to use a specified mass ratio.
      * </p>
      * @param primaryBody Primary Body in the CR3BP System
      * @param secondaryBody Secondary Body in the CR3BP System
      * @param a Semi-Major Axis of the secondary body
      * @param mu Mass ratio of the CR3BPSystem
      */
     public CR3BPSystem(final CelestialBody primaryBody, final CelestialBody secondaryBody, final double a, final double mu) {
         this.primaryBody = primaryBody;
         this.secondaryBody = secondaryBody;
 
         this.name = primaryBody.getName() + "_" + secondaryBody.getName();
 
         final double mu1 = primaryBody.getGM();
 
         this.mu = mu;
         this.dDim = a;
         this.vDim = FastMath.sqrt(mu1 / dDim);
         this.tDim = 2 * FastMath.PI * dDim / vDim;
         this.rotatingFrame = new CR3BPRotatingFrame(mu, primaryBody, secondaryBody);
 
         computeLagrangianPointsPosition();
 
         // Calculation of collinear points gamma
 
         // L1 Gamma
         final double x1 = l1Position.getX();
         final double dCP1 = 1 - mu;
         this.gamma1 = dCP1 - x1;
 
         // L2 Gamma
         final double x2 = l2Position.getX();
         final double dCP2 = 1 - mu;
         this.gamma2 = x2 - dCP2;
 
         // L3 Gamma
         final double x3 = l3Position.getX();
         final double dCP3 = -mu;
         this.gamma3 = dCP3 - x3;
 
     }
 
     /** Calculation of Lagrangian Points position using CR3BP equations.
      */
     private void computeLagrangianPointsPosition() {
         // Calculation of Lagrangian Points position using CR3BP equations
 
         // L1
         final BracketingNthOrderBrentSolver solver =
                         new BracketingNthOrderBrentSolver(RELATIVE_ACCURACY,
                                                           ABSOLUTE_ACCURACY,
                                                           FUNCTION_ACCURACY, MAX_ORDER);
 
         final double baseR1 = 1 - FastMath.cbrt(mu / 3);
         final UnivariateFunction l1Equation = x -> {
             final double leq1 =
                             x * (x + mu) * (x + mu) * (x + mu - 1) * (x + mu - 1);
             final double leq2 = (1 - mu) * (x + mu - 1) * (x + mu - 1);
             final double leq3 = mu * (x + mu) * (x + mu);
             return leq1 - leq2 + leq3;
         };
         final double[] searchInterval1 =
                         UnivariateSolverUtils.bracket(l1Equation, baseR1, -mu,
                                                       1 - mu, 1E-6, 1,
                                                       MAX_EVALUATIONS);
 
         final double r1 =
                         solver.solve(MAX_EVALUATIONS, l1Equation, searchInterval1[0],
                                      searchInterval1[1], AllowedSolution.ANY_SIDE);
 
         this.l1Position = new Vector3D(r1, 0.0, 0.0);
 
         // L2
         final double baseR2 = 1 + FastMath.cbrt(mu / 3);
         final UnivariateFunction l2Equation = x -> {
             final double leq21 =
                             x * (x + mu) * (x + mu) * (x + mu - 1) * (x + mu - 1);
             final double leq22 = (1 - mu) * (x + mu - 1) * (x + mu - 1);
             final double leq23 = mu * (x + mu) * (x + mu);
             return leq21 - leq22 - leq23;
         };
         final double[] searchInterval2 =
                         UnivariateSolverUtils.bracket(l2Equation, baseR2, 1 - mu, 2, 1E-6,
                                                       1, MAX_EVALUATIONS);
 
         final double r2 =
                         solver.solve(MAX_EVALUATIONS, l2Equation, searchInterval2[0],
                                      searchInterval2[1], AllowedSolution.ANY_SIDE);
 
         this.l2Position = new Vector3D(r2, 0.0, 0.0);
 
         // L3
         final double baseR3 = -(1 + 5 * mu / 12);
         final UnivariateFunction l3Equation = x -> {
             final double leq31 =
                             x * (x + mu) * (x + mu) * (x + mu - 1) * (x + mu - 1);
             final double leq32 = (1 - mu) * (x + mu - 1) * (x + mu - 1);
             final double leq33 = mu * (x + mu) * (x + mu);
             return leq31 + leq32 + leq33;
         };
         final double[] searchInterval3 =
                         UnivariateSolverUtils.bracket(l3Equation, baseR3, -2, -mu, 1E-6, 1,
                                                       MAX_EVALUATIONS);
 
         final double r3 =
                         solver.solve(MAX_EVALUATIONS, l3Equation, searchInterval3[0],
                                      searchInterval3[1], AllowedSolution.ANY_SIDE);
 
         this.l3Position = new Vector3D(r3, 0.0, 0.0);
 
         // L4
         this.l4Position = new Vector3D(0.5 - mu, FastMath.sqrt(3) / 2, 0);
 
         // L5
         this.l5Position = new Vector3D(0.5 - mu, -FastMath.sqrt(3) / 2, 0);
     }
 
     /** Get the CR3BP mass ratio of the system mu2/(mu1+mu2).
      * @return CR3BP mass ratio of the system mu2/(mu1+mu2)
      */
     public double getMassRatio() {
         return mu;
     }
 
     /** Get the CR3BP distance between the two bodies.
      * @return CR3BP distance between the two bodies(m)
      */
     public double getDdim() {
         return dDim;
     }
 
     /** Get the CR3BP orbital velocity of m2.
      * @return CR3BP orbital velocity of m2(m/s)
      */
     public double getVdim() {
         return vDim;
     }
 
     /** Get the CR3BP orbital period of m2 around m1.
      * @return CR3BP orbital period of m2 around m1(s)
      */
     public double getTdim() {
         return tDim;
     }
 
     /** Get the name of the CR3BP system.
      * @return name of the CR3BP system
      */
     public String getName() {
         return name;
     }
 
     /** Get the primary CelestialBody.
      * @return primary CelestialBody
      */
     public CelestialBody getPrimary() {
         return primaryBody;
     }
 
     /** Get the secondary CelestialBody.
      * @return secondary CelestialBody
      */
     public CelestialBody getSecondary() {
         return secondaryBody;
     }
 
     /** Get the CR3BP Rotating Frame.
      * @return CR3BP Rotating Frame
      */
     public Frame getRotatingFrame() {
         return rotatingFrame;
     }
 
     /** Get the position of the Lagrangian point in the CR3BP Rotating frame.
      * @param lagrangianPoint Lagrangian Point to consider
      * @return position of the Lagrangian point in the CR3BP Rotating frame (-)
      */
     public Vector3D getLPosition(final LagrangianPoints lagrangianPoint) {
 
         final Vector3D lPosition;
 
         switch (lagrangianPoint) {
 
             case L1:
                 lPosition = l1Position;
                 break;
 
             case L3:
                 lPosition = l3Position;
                 break;
 
             case L4:
                 lPosition = l4Position;
                 break;
 
             case L5:
                 lPosition = l5Position;
                 break;
 
             default:
                 lPosition = l2Position;
                 break;
 
         }
         return lPosition;
     }
 
     /**
      * Get the position of the Lagrangian point in the CR3BP Rotating frame.
      * @param lagrangianPoint Lagrangian Point to consider
      * @return Distance between a Lagrangian Point and its closest primary.
      */
     public double getGamma(final LagrangianPoints lagrangianPoint) {
 
         final double gamma;
 
         switch (lagrangianPoint) {
 
             case L1:
                 gamma = gamma1;
                 break;
 
             case L2:
                 gamma = gamma2;
                 break;
 
             case L3:
                 gamma = gamma3;
                 break;
 
             default:
                 gamma = 0;
         }
         return gamma;
     }
 
     /** Get the PVCoordinates from normalized units to standard units in an output frame.
      * @param pv0 Normalized PVCoordinates in the rotating frame
      * @param date Date of the transform
      * @param outputFrame Frame in which the output PVCoordinates will be
      * @return PVCoordinates in the output frame [m,m/s]
      */
     private PVCoordinates getRealPV(final PVCoordinates pv0, final AbsoluteDate date, final Frame outputFrame) {
         // 1.   Dimensionalize  the  primary-centered  rotating  state  using  the  instantaneously
         //      defined characteristic quantities
         // 2.   Apply the transformation to primary inertial frame
         // 3.   Apply the transformation to output frame
 
         final Frame primaryInertialFrame = primaryBody.getInertiallyOrientedFrame();
         final Vector3D pv21 = secondaryBody.getPosition(date, primaryInertialFrame);
 
         // Distance and Velocity to dimensionalize the state vector
         final double dist12 = pv21.getNorm();
         final double vCircular  = FastMath.sqrt(primaryBody.getGM() / dist12);
 
         // Dimensionalized state vector centered on primary body
         final PVCoordinates pvDim = new PVCoordinates(pv0.getPosition().scalarMultiply(dist12),
                                                       pv0.getVelocity().scalarMultiply(vCircular));
 
         // Transformation between rotating frame and the primary inertial
         final Transform rotatingToPrimaryInertial = rotatingFrame.getTransformTo(primaryInertialFrame, date);
 
         // State vector in the primary inertial frame
         final PVCoordinates pv2 = rotatingToPrimaryInertial.transformPVCoordinates(pvDim);
 
 
         // Transformation between primary inertial frame and the output frame
         final Transform primaryInertialToOutputFrame = primaryInertialFrame.getTransformTo(outputFrame, date);
 
         return primaryInertialToOutputFrame.transformPVCoordinates(pv2);
     }
 
     /** Get the AbsolutePVCoordinates from normalized units to standard units in an output frame.
      * This method ensure the constituency of the date of returned AbsolutePVCoordinate, especially
      * when apv0 is the result of a propagation in CR3BP normalized model.
      * @param apv0 Normalized AbsolutePVCoordinates in the rotating frame
      * @param initialDate Date of the at the beginning of the propagation
      * @param outputFrame Frame in which the output AbsolutePVCoordinates will be
      * @return AbsolutePVCoordinates in the output frame [m,m/s]
      */
     public AbsolutePVCoordinates getRealAPV(final AbsolutePVCoordinates apv0, final AbsoluteDate initialDate, final Frame outputFrame) {
 
         final double duration = apv0.getDate().durationFrom(initialDate) * tDim / (2 * FastMath.PI);
         final AbsoluteDate date = initialDate.shiftedBy(duration);
 
         // PVCoordinate in the output frame
         final PVCoordinates pv3 = getRealPV(apv0.getPVCoordinates(), date, outputFrame);
 
         return new AbsolutePVCoordinates(outputFrame, date, pv3);
     }
 
 }