StateCovariance.java

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package org.orekit.propagation;

import org.hipparchus.analysis.differentiation.Gradient;
import org.hipparchus.analysis.differentiation.GradientField;
import org.hipparchus.linear.Array2DRowRealMatrix;
import org.hipparchus.linear.MatrixUtils;
import org.hipparchus.linear.RealMatrix;
import org.hipparchus.util.MathArrays;
import org.orekit.errors.OrekitException;
import org.orekit.errors.OrekitMessages;
import org.orekit.frames.Frame;
import org.orekit.frames.KinematicTransform;
import org.orekit.frames.LOF;
import org.orekit.orbits.FieldOrbit;
import org.orekit.orbits.Orbit;
import org.orekit.orbits.OrbitType;
import org.orekit.orbits.PositionAngleType;
import org.orekit.time.AbsoluteDate;
import org.orekit.time.FieldAbsoluteDate;
import org.orekit.time.TimeStamped;
import org.orekit.utils.CartesianCovarianceUtils;

/** This class is the representation of a covariance matrix at a given date.
 * <p>
 * Currently, the covariance only represents the orbital elements.
 * <p>
 * It is possible to change the covariance frame by using the
 * {@link #changeCovarianceFrame(Orbit, Frame)} or {@link #changeCovarianceFrame(Orbit, LOF)} method.
 * These methods are based on Equations (18) and (20) of <i>Covariance Transformations for Satellite
 * Flight Dynamics Operations</i> by David A. SVallado.
 * <p>
 * Finally, covariance orbit type can be changed using the
 * {@link #changeCovarianceType(Orbit, OrbitType, PositionAngleType)} method.
 *
 * @author Bryan Cazabonne
 * @author Vincent Cucchietti
 * @since 11.3
 */
public class StateCovariance implements TimeStamped {

    /** State dimension. */
    public static final int STATE_DIMENSION = 6;

    /** Default position angle for covariance expressed in Cartesian elements. */
    private static final PositionAngleType DEFAULT_POSITION_ANGLE = PositionAngleType.TRUE;

    /** Orbital covariance [6x6]. */
    private final RealMatrix orbitalCovariance;

    /** Covariance frame (can be null if LOF is defined). */
    private final Frame frame;

    /** Covariance LOF type (can be null if frame is defined). */
    private final LOF lof;

    /** Covariance epoch. */
    private final AbsoluteDate epoch;

    /** Covariance orbit type. */
    private final OrbitType orbitType;

    /** Covariance position angle type (not used if orbitType is CARTESIAN). */
    private final PositionAngleType angleType;

    /**
     * Constructor.
     * @param orbitalCovariance 6x6 orbital parameters covariance
     * @param epoch epoch of the covariance
     * @param lof covariance LOF type
     */
    public StateCovariance(final RealMatrix orbitalCovariance, final AbsoluteDate epoch, final LOF lof) {
        this(orbitalCovariance, epoch, null, lof, OrbitType.CARTESIAN, DEFAULT_POSITION_ANGLE);
    }

    /**
     * Constructor.
     * @param orbitalCovariance 6x6 orbital parameters covariance
     * @param epoch epoch of the covariance
     * @param covarianceFrame covariance frame (inertial or Earth fixed)
     * @param orbitType orbit type of the covariance (CARTESIAN if covarianceFrame is not pseudo-inertial)
     * @param angleType position angle type of the covariance (not used if orbitType is CARTESIAN)
     */
    public StateCovariance(final RealMatrix orbitalCovariance, final AbsoluteDate epoch,
                           final Frame covarianceFrame,
                           final OrbitType orbitType, final PositionAngleType angleType) {
        this(orbitalCovariance, epoch, covarianceFrame, null, orbitType, angleType);
    }

    /**
     * Private constructor.
     * @param orbitalCovariance 6x6 orbital parameters covariance
     * @param epoch epoch of the covariance
     * @param covarianceFrame covariance frame (inertial or Earth fixed)
     * @param lof covariance LOF type
     * @param orbitType orbit type of the covariance
     * @param angleType position angle type of the covariance (not used if orbitType is CARTESIAN)
     */
    private StateCovariance(final RealMatrix orbitalCovariance, final AbsoluteDate epoch,
                            final Frame covarianceFrame, final LOF lof,
                            final OrbitType orbitType, final PositionAngleType angleType) {

        checkFrameAndOrbitTypeConsistency(covarianceFrame, orbitType);

        this.orbitalCovariance = orbitalCovariance;
        this.epoch = epoch;
        this.frame     = covarianceFrame;
        this.lof       = lof;
        this.orbitType = orbitType;
        this.angleType = angleType;

    }

    /**
     * Check constructor's inputs consistency.
     *
     * @param covarianceFrame covariance frame (inertial or Earth fixed)
     * @param inputType orbit type of the covariance
     *
     * @throws OrekitException if input frame is not pseudo-inertial AND the orbit type is not Cartesian
     */
    public static void checkFrameAndOrbitTypeConsistency(final Frame covarianceFrame, final OrbitType inputType) {

        // State covariance expressed in a celestial body frame
        if (covarianceFrame != null) {

            // Input frame is not pseudo-inertial
            if (!covarianceFrame.isPseudoInertial() && inputType != OrbitType.CARTESIAN) {
                throw new OrekitException(OrekitMessages.WRONG_ORBIT_PARAMETERS_TYPE,
                                          inputType.name(),
                                          OrbitType.CARTESIAN.name());
            }
        }
    }

    /**
     * Checks if input/output orbit and angle types are identical.
     *
     * @param inOrbitType input orbit type
     * @param inAngleType input angle type
     * @param outOrbitType output orbit type
     * @param outAngleType output angle type
     * @return flag defining if input/output orbit and angle types are identical
     */
    public static boolean inputAndOutputAreIdentical(final OrbitType inOrbitType, final PositionAngleType inAngleType,
                                                     final OrbitType outOrbitType, final PositionAngleType outAngleType) {
        return inOrbitType == outOrbitType && inAngleType == outAngleType;
    }

    /**
     * Checks if input and output orbit types are both {@code OrbitType.CARTESIAN}.
     *
     * @param inOrbitType input orbit type
     * @param outOrbitType output orbit type
     * @return flag defining if input and output orbit types are both {@code OrbitType.CARTESIAN}
     */
    public static boolean inputAndOutputOrbitTypesAreCartesian(final OrbitType inOrbitType, final OrbitType outOrbitType) {
        return inOrbitType == OrbitType.CARTESIAN && outOrbitType == OrbitType.CARTESIAN;
    }

    /** {@inheritDoc}. */
    @Override
    public AbsoluteDate getDate() {
        return epoch;
    }

    /**
     * Get the covariance matrix.
     * @return the covariance matrix
     */
    public RealMatrix getMatrix() {
        return orbitalCovariance;
    }

    /**
     * Get the covariance orbit type.
     * @return the covariance orbit type
     */
    public OrbitType getOrbitType() {
        return orbitType;
    }

    /**
     * Get the covariance angle type.
     * @return the covariance angle type
     */
    public PositionAngleType getPositionAngleType() {
        return angleType;
    }

    /**
     * Get the covariance frame.
     * @return the covariance frame (can be null)
     * @see #getLOF()
     */
    public Frame getFrame() {
        return frame;
    }

    /**
     * Get the covariance LOF type.
     * @return the covariance LOF type (can be null)
     * @see #getFrame()
     */
    public LOF getLOF() {
        return lof;
    }

    /**
     * Get the covariance matrix in another orbit type.
     * <p>
     * The covariance orbit type <b>cannot</b> be changed if the covariance
     * matrix is expressed in a {@link LOF local orbital frame} or a
     * non-pseudo inertial frame.
     * <p>
     * As this type change uses the jacobian matrix of the transformation, it introduces a linear approximation.
     * Hence, the current covariance matrix <b>will not exactly match</b> the new linearized case and the
     * distribution will not follow a generalized Gaussian distribution anymore.
     * <p>
     * This is based on equation (1) to (6) from "Vallado, D. A. (2004). Covariance transformations for satellite flight
     * dynamics operations."
     *
     * @param orbit orbit to which the covariance matrix should correspond
     * @param outOrbitType target orbit type of the state covariance matrix
     * @param outAngleType target position angle type of the state covariance matrix
     * @return a new covariance state, expressed in the target orbit type with the target position angle
     * @see #changeCovarianceFrame(Orbit, Frame)
     */
    public StateCovariance changeCovarianceType(final Orbit orbit, final OrbitType outOrbitType,
                                                final PositionAngleType outAngleType) {

        // Handle case where the covariance is already expressed in the output type
        if (outOrbitType == orbitType && (outOrbitType == OrbitType.CARTESIAN || outAngleType == angleType)) {
            if (lof == null) {
                return new StateCovariance(orbitalCovariance, epoch, frame, orbitType, angleType);
            }
            else {
                return new StateCovariance(orbitalCovariance, epoch, lof);
            }
        }

        // Check if the covariance is expressed in a celestial body frame
        if (frame != null) {

            // Check if the covariance is defined in an inertial frame
            if (frame.isPseudoInertial()) {
                return changeTypeAndCreate(orbit, epoch, frame, orbitType, angleType, outOrbitType, outAngleType,
                                           orbitalCovariance);
            }

            // The covariance is not defined in an inertial frame. The orbit type cannot be changed
            throw new OrekitException(OrekitMessages.CANNOT_CHANGE_COVARIANCE_TYPE_IF_DEFINED_IN_NON_INERTIAL_FRAME);

        }

        // The covariance is not expressed in a celestial body frame. The orbit type cannot be changed
        throw new OrekitException(OrekitMessages.CANNOT_CHANGE_COVARIANCE_TYPE_IF_DEFINED_IN_LOF);

    }

    /**
     * Get the covariance in a given local orbital frame.
     * <p>
     * Changing the covariance frame is a linear process, this method does not introduce approximation unless a change
     * in covariance orbit type is required.
     * <p>
     * This is based on equation (18) to (20) "from Vallado, D. A. (2004). Covariance transformations for satellite
     * flight dynamics operations."
     *
     * @param orbit orbit to which the covariance matrix should correspond
     * @param lofOut output local orbital frame
     * @return a new covariance state, expressed in the output local orbital frame
     */
    public StateCovariance changeCovarianceFrame(final Orbit orbit, final LOF lofOut) {

        // Verify current covariance frame
        if (lof != null) {

            // Check specific case where output covariance will be the same
            if (lofOut == lof) {
                return new StateCovariance(orbitalCovariance, epoch, lof);
            }

            // Change the covariance local orbital frame
            return changeFrameAndCreate(orbit, epoch, lof, lofOut, orbitalCovariance);

        } else {

            // Covariance is expressed in celestial body frame
            return changeFrameAndCreate(orbit, epoch, frame, lofOut, orbitalCovariance, orbitType, angleType);

        }

    }

    /**
     * Get the covariance in the output frame.
     * <p>
     * Changing the covariance frame is a linear process, this method does not introduce approximation unless a change
     * in covariance orbit type is required.
     * <p>
     * This is based on equation (18) to (20) "from Vallado, D. A. (2004). Covariance transformations for satellite
     * flight dynamics operations."
     *
     * @param orbit orbit to which the covariance matrix should correspond
     * @param frameOut output frame
     * @return a new covariance state, expressed in the output frame
     */
    public StateCovariance changeCovarianceFrame(final Orbit orbit, final Frame frameOut) {

        // Verify current covariance frame
        if (lof != null) {

            // Covariance is expressed in local orbital frame
            return changeFrameAndCreate(orbit, epoch, lof, frameOut, orbitalCovariance);

        } else {

            // Check specific case where output covariance will be the same
            if (frame == frameOut) {
                return new StateCovariance(orbitalCovariance, epoch, frame, orbitType, angleType);
            }

            // Change covariance frame
            return changeFrameAndCreate(orbit, epoch, frame, frameOut, orbitalCovariance, orbitType, angleType);

        }

    }

    /**
     * Get a time-shifted covariance matrix.
     * <p>
     * The shifting model is a linearized, Keplerian one. In other words, it is based on a state transition matrix that
     * is computed assuming Keplerian motion.
     * <p>
     * Shifting is <em>not</em> intended as a replacement for proper covariance propagation, but should be sufficient
     * for small time shifts or coarse accuracy.
     *
     * @param orbit orbit to which the covariance matrix should correspond
     * @param dt time shift in seconds
     * @return a new covariance state, shifted with respect to the instance
     */
    public StateCovariance shiftedBy(final Orbit orbit, final double dt) {

        // Shifted orbit
        final Orbit shifted = orbit.shiftedBy(dt);

        // State covariance expressed in celestial body frame
        if (frame != null) {

            // State covariance expressed in a pseudo-inertial frame
            if (frame.isPseudoInertial()) {

                // Shift covariance by applying the STM
                final RealMatrix stm = getKeplerianStm(orbit, dt);
                return new StateCovariance(stm.multiply(getMatrix().multiplyTransposed(stm)), getDate().shiftedBy(dt),
                        frame, orbitType, getPositionAngleType());

            }

            // State covariance expressed in a non pseudo-inertial frame
            else {

                // Convert state covariance to orbit pseudo-inertial frame
                final StateCovariance inOrbitFrame = changeCovarianceFrame(orbit, orbit.getFrame());

                // Shift the state covariance
                final StateCovariance shiftedCovariance = inOrbitFrame.shiftedBy(orbit, dt);

                // Restore the initial covariance frame
                return shiftedCovariance.changeCovarianceFrame(shifted, frame);
            }
        }

        // State covariance expressed in a commonly used local orbital frame (LOF)
        else {

            // Convert state covariance to orbit pseudo-inertial frame
            final StateCovariance inOrbitFrame = changeCovarianceFrame(orbit, orbit.getFrame());

            // Shift the state covariance
            final StateCovariance shiftedCovariance = inOrbitFrame.shiftedBy(orbit, dt);

            // Restore the initial covariance frame
            return shiftedCovariance.changeCovarianceFrame(shifted, lof);
        }

    }

    /**
     * Create a covariance matrix in another orbit type.
     * <p>
     * As this type change uses the jacobian matrix of the transformation, it introduces a linear approximation.
     * Hence, the input covariance matrix <b>will not exactly match</b> the new linearized case and the
     * distribution will not follow a generalized Gaussian distribution anymore.
     * <p>
     * This is based on equation (1) to (6) from "Vallado, D. A. (2004). Covariance transformations for satellite flight
     * dynamics operations."
     *
     * @param orbit orbit to which the covariance matrix should correspond
     * @param date covariance epoch
     * @param covFrame covariance frame
     * @param inOrbitType initial orbit type of the state covariance matrix
     * @param inAngleType initial position angle type of the state covariance matrix
     * @param outOrbitType target orbit type of the state covariance matrix
     * @param outAngleType target position angle type of the state covariance matrix
     * @param inputCov input covariance
     * @return the covariance expressed in the target orbit type with the target position angle
     */
    private static StateCovariance changeTypeAndCreate(final Orbit orbit,
                                                       final AbsoluteDate date,
                                                       final Frame covFrame,
                                                       final OrbitType inOrbitType, final PositionAngleType inAngleType,
                                                       final OrbitType outOrbitType, final PositionAngleType outAngleType,
                                                       final RealMatrix inputCov) {

        // Check if type change is really necessary, if not then return input covariance
        if (inputAndOutputOrbitTypesAreCartesian(inOrbitType, outOrbitType) ||
            inputAndOutputAreIdentical(inOrbitType, inAngleType, outOrbitType, outAngleType)) {
            return new StateCovariance(inputCov, date, covFrame, inOrbitType, inAngleType);
        }

        // Notations:
        // I: Input orbit type
        // O: Output orbit type
        // C: Cartesian parameters

        // Compute dOutputdCartesian
        final double[][] aOC               = new double[STATE_DIMENSION][STATE_DIMENSION];
        final Orbit      orbitInOutputType = outOrbitType.convertType(orbit);
        orbitInOutputType.getJacobianWrtCartesian(outAngleType, aOC);
        final RealMatrix dOdC = new Array2DRowRealMatrix(aOC, false);

        // Compute dCartesiandInput
        final double[][] aCI              = new double[STATE_DIMENSION][STATE_DIMENSION];
        final Orbit      orbitInInputType = inOrbitType.convertType(orbit);
        orbitInInputType.getJacobianWrtParameters(inAngleType, aCI);
        final RealMatrix dCdI = new Array2DRowRealMatrix(aCI, false);

        // Compute dOutputdInput
        final RealMatrix dOdI = dOdC.multiply(dCdI);

        // Conversion of the state covariance matrix in target orbit type
        final RealMatrix outputCov = dOdI.multiply(inputCov.multiplyTransposed(dOdI));

        // Return the converted covariance
        return new StateCovariance(outputCov, date, covFrame, outOrbitType, outAngleType);

    }

    /**
     * Create a covariance matrix from a {@link LOF local orbital frame} to another
     * {@link LOF local orbital frame}.
     * <p>
     * Changing the covariance frame is a linear process, this method does not introduce approximation.
     * <p>
     * The transformation is based on Equation (18) to (20) of "Covariance Transformations for Satellite Flight Dynamics
     * Operations" by David A. Vallado.
     *
     * @param orbit orbit to which the covariance matrix should correspond
     * @param date covariance epoch
     * @param lofIn the local orbital frame in which the input covariance matrix is expressed
     * @param lofOut the target local orbital frame
     * @param inputCartesianCov input covariance {@code CARTESIAN})
     * @return the covariance matrix expressed in the target commonly used local orbital frame in Cartesian elements
     */
    private static StateCovariance changeFrameAndCreate(final Orbit orbit, final AbsoluteDate date,
                                                        final LOF lofIn, final LOF lofOut,
                                                        final RealMatrix inputCartesianCov) {

        // Builds the matrix to perform covariance transformation
        final RealMatrix jacobianFromLofInToLofOut =
                getJacobian(LOF.transformFromLOFInToLOFOut(lofIn, lofOut, date, orbit.getPVCoordinates()));

        // Get the Cartesian covariance matrix converted to frameOut
        final RealMatrix cartesianCovarianceOut =
                jacobianFromLofInToLofOut.multiply(inputCartesianCov.multiplyTransposed(jacobianFromLofInToLofOut));

        // Output converted covariance
        return new StateCovariance(cartesianCovarianceOut, date, lofOut);

    }

    /**
     * Convert the covariance matrix from a {@link Frame frame} to a {@link LOF local orbital frame}.
     * <p>
     * Changing the covariance frame is a linear process, this method does not introduce approximation unless a change
     * in covariance orbit type is required.
     * <p>
     * The transformation is based on Equation (18) to (20) of "Covariance Transformations for Satellite Flight Dynamics
     * Operations" by David A. Vallado.
     * <p>
     * As the frame transformation must be performed with the covariance expressed in Cartesian elements, both the orbit
     * and position angle types of the input covariance must be provided.
     * <p>
     * <b>The output covariance matrix will provided in Cartesian orbit type and not converted back to
     * its original expression (if input different from Cartesian elements).</b>
     *
     * @param orbit orbit to which the covariance matrix should correspond
     * @param date covariance epoch
     * @param frameIn the frame in which the input covariance matrix is expressed. In case the frame is <b>not</b>
     * pseudo-inertial, the input covariance matrix is expected to be expressed in <b>Cartesian elements</b>.
     * @param lofOut the target local orbital frame
     * @param inputCov input covariance
     * @param covOrbitType orbit type of the covariance matrix (used if frameIn is pseudo-inertial)
     * @param covAngleType position angle type of the covariance matrix (used if frameIn is pseudo-inertial) (not used
     * if covOrbitType equals {@code CARTESIAN})
     * @return the covariance matrix expressed in the target local orbital frame in Cartesian elements
     * @throws OrekitException if input frame is <b>not</b> pseudo-inertial <b>and</b> the input covariance is
     * <b>not</b> expressed in Cartesian elements.
     */
    private static StateCovariance changeFrameAndCreate(final Orbit orbit, final AbsoluteDate date,
                                                        final Frame frameIn, final LOF lofOut,
                                                        final RealMatrix inputCov,
                                                        final OrbitType covOrbitType,
                                                        final PositionAngleType covAngleType) {

        // Input frame is inertial
        if (frameIn.isPseudoInertial()) {

            // Convert input matrix to Cartesian parameters in input frame
            final RealMatrix cartesianCovarianceIn =
                    changeTypeAndCreate(orbit, date, frameIn, covOrbitType, covAngleType,
                                        OrbitType.CARTESIAN, PositionAngleType.MEAN,
                                        inputCov).getMatrix();

            // Builds the matrix to perform covariance transformation
            final RealMatrix jacobianFromFrameInToLofOut =
                            getJacobian(lofOut.transformFromInertial(date, orbit.getPVCoordinates(frameIn)));

            // Get the Cartesian covariance matrix converted to frameOut
            final RealMatrix cartesianCovarianceOut =
                    jacobianFromFrameInToLofOut.multiply(cartesianCovarianceIn.multiplyTransposed(jacobianFromFrameInToLofOut));

            // Return converted covariance matrix expressed in Cartesian elements
            return new StateCovariance(cartesianCovarianceOut, date, lofOut);

        }

        // Input frame is not inertial so the covariance matrix is expected to be in Cartesian elements
        else {

            // Method checkInputConsistency already verify that input covariance is defined in CARTESIAN type
            final Frame orbitInertialFrame = orbit.getFrame();

            // Compute rotation matrix from frameIn to orbit inertial frame
            final RealMatrix cartesianCovarianceInOrbitFrame =
                   changeFrameAndCreate(orbit, date, frameIn, orbitInertialFrame, inputCov,
                                         OrbitType.CARTESIAN, PositionAngleType.MEAN).getMatrix();

            // Convert from orbit inertial frame to lofOut
            return changeFrameAndCreate(orbit, date, orbitInertialFrame, lofOut, cartesianCovarianceInOrbitFrame,
                                        OrbitType.CARTESIAN, PositionAngleType.MEAN);

        }

    }

    /**
     * Convert the covariance matrix from a {@link LOF  local orbital frame} to a {@link Frame frame}.
     * <p>
     * Changing the covariance frame is a linear process, this method does not introduce approximation.
     * <p>
     * The transformation is based on Equation (18) to (20) of "Covariance Transformations for Satellite Flight Dynamics
     * Operations" by David A. Vallado.
     * <p>
     * The <u>input</u> covariance matrix is necessarily provided in <b>Cartesian orbit type</b>.
     * <p>
     * The <u>output</u> covariance matrix will be expressed in <b>Cartesian elements</b>.
     *
     * @param orbit orbit to which the covariance matrix should correspond
     * @param date covariance epoch
     * @param lofIn the local orbital frame in which the input covariance matrix is expressed
     * @param frameOut the target frame
     * @param inputCartesianCov input covariance ({@code CARTESIAN})
     * @return the covariance matrix expressed in the target frame in Cartesian elements
     */
    private static StateCovariance changeFrameAndCreate(final Orbit orbit, final AbsoluteDate date,
                                                        final LOF lofIn, final Frame frameOut,
                                                        final RealMatrix inputCartesianCov) {

        // Output frame is pseudo-inertial
        if (frameOut.isPseudoInertial()) {

            // Builds the matrix to perform covariance transformation
            final RealMatrix jacobianFromLofInToFrameOut =
                    getJacobian(lofIn.transformFromInertial(date, orbit.getPVCoordinates(frameOut)).getInverse());

            // Transform covariance
            final RealMatrix transformedCovariance =
                    jacobianFromLofInToFrameOut.multiply(inputCartesianCov.multiplyTransposed(jacobianFromLofInToFrameOut));

            // Get the Cartesian covariance matrix converted to frameOut
            return new StateCovariance(transformedCovariance, date, frameOut, OrbitType.CARTESIAN,
                                       DEFAULT_POSITION_ANGLE);

        }

        // Output frame is not pseudo-inertial
        else {

            // Builds the matrix to perform covariance transformation
            final RealMatrix jacobianFromLofInToOrbitFrame =
                    getJacobian(lofIn.transformFromInertial(date, orbit.getPVCoordinates()).getInverse());

            // Get the Cartesian covariance matrix converted to orbit inertial frame
            final RealMatrix cartesianCovarianceInOrbitFrame = jacobianFromLofInToOrbitFrame.multiply(
                    inputCartesianCov.multiplyTransposed(jacobianFromLofInToOrbitFrame));

            // Get the Cartesian covariance matrix converted to frameOut
            return changeFrameAndCreate(orbit, date, orbit.getFrame(), frameOut, cartesianCovarianceInOrbitFrame,
                                        OrbitType.CARTESIAN, PositionAngleType.MEAN);
        }

    }

    /**
     * Get the covariance matrix in another frame.
     * <p>
     * The transformation is based on Equation (18) of "Covariance Transformations for Satellite Flight Dynamics
     * Operations" by David A. Vallado.
     * <p>
     * Changing the covariance frame is a linear process, this method does not introduce approximation unless a change
     * in covariance orbit type is required.
     * <p>
     * As the frame transformation must be performed with the covariance expressed in Cartesian elements, both the orbit
     * and position angle types of the input covariance must be provided.
     * <p>
     * In case the <u>input</u> frame is <b>not</b> pseudo-inertial, the <u>input</u> covariance matrix is necessarily
     * expressed in <b>Cartesian elements</b>.
     * <p>
     * In case the <u>output</u> frame is <b>not</b> pseudo-inertial, the <u>output</u> covariance matrix will be
     * expressed in <b>Cartesian elements</b>.
     *
     * @param orbit orbit to which the covariance matrix should correspond
     * @param date covariance epoch
     * @param frameIn the frame in which the input covariance matrix is expressed
     * @param frameOut the target frame
     * @param inputCov input covariance
     * @param covOrbitType orbit type of the covariance matrix (used if frameIn is pseudo-inertial)
     * @param covAngleType position angle type of the covariance matrix (used if frameIn is pseudo-inertial) (<b>not</b>
     * used if covOrbitType equals {@code CARTESIAN})
     * @return the covariance matrix expressed in the target frame
     * @throws OrekitException if input frame is <b>not</b> pseudo-inertial <b>and</b> the input covariance is
     * <b>not</b> expressed in Cartesian elements.
     */
    private static StateCovariance changeFrameAndCreate(final Orbit orbit, final AbsoluteDate date,
                                                        final Frame frameIn, final Frame frameOut,
                                                        final RealMatrix inputCov,
                                                        final OrbitType covOrbitType,
                                                        final PositionAngleType covAngleType) {

        // Input frame pseudo-inertial
        if (frameIn.isPseudoInertial()) {

            // Convert input matrix to Cartesian parameters in input frame
            final RealMatrix cartesianCovarianceIn =
                    changeTypeAndCreate(orbit, date, frameIn, covOrbitType, covAngleType,
                                        OrbitType.CARTESIAN, PositionAngleType.MEAN,
                                        inputCov).getMatrix();

            // Get the Cartesian covariance matrix converted to frameOut
            final RealMatrix cartesianCovarianceOut = CartesianCovarianceUtils.changeReferenceFrame(frameIn,
                    cartesianCovarianceIn, date, frameOut);

            // Output frame is pseudo-inertial
            if (frameOut.isPseudoInertial()) {

                // Convert output Cartesian matrix to initial orbit type and position angle
                return changeTypeAndCreate(orbit, date, frameOut, OrbitType.CARTESIAN, PositionAngleType.MEAN,
                                           covOrbitType, covAngleType, cartesianCovarianceOut);

            }

            // Output frame is not pseudo-inertial
            else {

                // Output Cartesian matrix
                return new StateCovariance(cartesianCovarianceOut, date, frameOut, OrbitType.CARTESIAN,
                                           DEFAULT_POSITION_ANGLE);

            }
        }

        // Input frame is not pseudo-inertial
        else {

            // Method checkInputConsistency already verify that input covariance is defined in CARTESIAN type

            // Convert covariance matrix to frameOut
            final RealMatrix covInFrameOut = CartesianCovarianceUtils.changeReferenceFrame(frameIn,
                    inputCov, date, frameOut);

            // Output the Cartesian covariance matrix converted to frameOut
            return new StateCovariance(covInFrameOut, date, frameOut, OrbitType.CARTESIAN, DEFAULT_POSITION_ANGLE);

        }

    }

    /**
     * Builds the matrix to perform covariance frame transformation.
     * @param transform input transformation
     * @return the matrix to perform the covariance frame transformation
     */
    private static RealMatrix getJacobian(final KinematicTransform transform) {
        return MatrixUtils.createRealMatrix(transform.getPVJacobian());
    }

    /**
     * Get the state transition matrix considering Keplerian contribution only and assuming equinoctial elements with mean anomaly.
     *
     * @param initialOrbit orbit to which the initial covariance matrix should correspond
     * @param dt time difference between the two orbits
     * @return the state transition matrix used to shift the covariance matrix
     * @deprecated since 13.1. If you must, do:
     * final RealMatrix stm = MatrixUtils.createRealIdentityMatrix(STATE_DIMENSION);
     * double contribution = initialOrbit.getMeanAnomalyDotWrtA() * dt;
     * stm.setEntry(5, 0, contribution);
     */
    @Deprecated
    public static RealMatrix getStm(final Orbit initialOrbit, final double dt) {
        // initialize the STM
        final RealMatrix stm = MatrixUtils.createRealIdentityMatrix(STATE_DIMENSION);

        // State transition matrix using Keplerian contribution only
        final double contribution = initialOrbit.getMeanAnomalyDotWrtA() * dt;
        stm.setEntry(5, 0, contribution);

        // Return
        return stm;
    }

    /**
     * Get the state transition matrix of Keplerian motion.
     * The coordinates used are those of the covariance itself.
     *
     * @param initialOrbit orbit to which the initial covariance matrix should correspond
     * @param dt time difference between the two orbits
     * @return the state transition matrix used to shift the covariance matrix
     * @since 13.1
     */
    RealMatrix getKeplerianStm(final Orbit initialOrbit, final double dt) {

        // compute derivatives of Keplerian motion
        final GradientField field = GradientField.getField(STATE_DIMENSION);
        final double[] stateVector = new double[STATE_DIMENSION];
        orbitType.mapOrbitToArray(initialOrbit, getPositionAngleType(), stateVector, null);
        final Gradient[] fieldGradientStateVector = MathArrays.buildArray(field, STATE_DIMENSION);
        for (int i = 0; i < STATE_DIMENSION; i++) {
            fieldGradientStateVector[i] = Gradient.variable(STATE_DIMENSION, i, stateVector[i]);
        }
        final FieldAbsoluteDate<Gradient> fieldGradientDate = new FieldAbsoluteDate<>(field, initialOrbit.getDate());
        final FieldOrbit<Gradient> fieldOrbit = orbitType.mapArrayToOrbit(fieldGradientStateVector, null, getPositionAngleType(),
                fieldGradientDate, field.getOne().newInstance(initialOrbit.getMu()), initialOrbit.getFrame());
        final FieldOrbit<Gradient> shiftedOrbit = fieldOrbit.shiftedBy(dt);  // automatic differentiation
        final Gradient[] gradient = MathArrays.buildArray(field, STATE_DIMENSION);
        orbitType.mapOrbitToArray(shiftedOrbit, getPositionAngleType(), gradient, null);

        // Return state transition matrix
        final RealMatrix stm = MatrixUtils.createRealIdentityMatrix(STATE_DIMENSION);
        for (int i = 0; i < gradient.length; i++) {
            stm.setRow(i, gradient[i].getGradient());
        }
        return stm;

    }

}