1   /* Copyright 2002-2024 CS GROUP
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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
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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
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16   */
17  package org.orekit.utils;
18  
19  import java.io.Serializable;
20  
21  import org.hipparchus.analysis.differentiation.DSFactory;
22  import org.hipparchus.analysis.differentiation.Derivative;
23  import org.hipparchus.analysis.differentiation.DerivativeStructure;
24  import org.hipparchus.analysis.differentiation.UnivariateDerivative1;
25  import org.hipparchus.analysis.differentiation.UnivariateDerivative2;
26  import org.hipparchus.exception.MathIllegalArgumentException;
27  import org.hipparchus.geometry.euclidean.threed.FieldVector3D;
28  import org.hipparchus.geometry.euclidean.threed.Vector3D;
29  import org.hipparchus.util.Blendable;
30  import org.hipparchus.util.FastMath;
31  import org.orekit.errors.OrekitException;
32  import org.orekit.errors.OrekitMessages;
33  import org.orekit.time.TimeShiftable;
34  
35  /** Simple container for Position/Velocity/Acceleration triplets.
36   * <p>
37   * The state can be slightly shifted to close dates. This shift is based on
38   * a simple quadratic model. It is <em>not</em> intended as a replacement for
39   * proper orbit propagation (it is not even Keplerian!) but should be sufficient
40   * for either small time shifts or coarse accuracy.
41   * </p>
42   * <p>
43   * This class is the angular counterpart to {@link AngularCoordinates}.
44   * </p>
45   * <p>Instances of this class are guaranteed to be immutable.</p>
46   * @author Fabien Maussion
47   * @author Luc Maisonobe
48   */
49  public class PVCoordinates implements TimeShiftable<PVCoordinates>, Blendable<PVCoordinates>, Serializable {
50  
51      /** Fixed position/velocity at origin (both p, v and a are zero vectors). */
52      public static final PVCoordinates ZERO = new PVCoordinates(Vector3D.ZERO, Vector3D.ZERO, Vector3D.ZERO);
53  
54      /** Serializable UID. */
55      private static final long serialVersionUID = 20140407L;
56  
57      /** The position. */
58      private final Vector3D position;
59  
60      /** The velocity. */
61      private final Vector3D velocity;
62  
63      /** The acceleration. */
64      private final Vector3D acceleration;
65  
66      /** Simple constructor.
67       * <p> Set the Coordinates to default : (0 0 0), (0 0 0), (0 0 0).</p>
68       */
69      public PVCoordinates() {
70          position     = Vector3D.ZERO;
71          velocity     = Vector3D.ZERO;
72          acceleration = Vector3D.ZERO;
73      }
74  
75      /** Builds a PVCoordinates triplet with zero acceleration.
76       * <p>Acceleration is set to zero</p>
77       * @param position the position vector (m)
78       * @param velocity the velocity vector (m/s)
79       */
80      public PVCoordinates(final Vector3D position, final Vector3D velocity) {
81          this.position     = position;
82          this.velocity     = velocity;
83          this.acceleration = Vector3D.ZERO;
84      }
85  
86      /** Builds a PVCoordinates triplet.
87       * @param position the position vector (m)
88       * @param velocity the velocity vector (m/s)
89       * @param acceleration the acceleration vector (m/s²)
90       */
91      public PVCoordinates(final Vector3D position, final Vector3D velocity, final Vector3D acceleration) {
92          this.position     = position;
93          this.velocity     = velocity;
94          this.acceleration = acceleration;
95      }
96  
97      /** Multiplicative constructor.
98       * <p>Build a PVCoordinates from another one and a scale factor.</p>
99       * <p>The PVCoordinates built will be a * pv</p>
100      * @param a scale factor
101      * @param pv base (unscaled) PVCoordinates
102      */
103     public PVCoordinates(final double a, final PVCoordinates pv) {
104         position     = new Vector3D(a, pv.position);
105         velocity     = new Vector3D(a, pv.velocity);
106         acceleration = new Vector3D(a, pv.acceleration);
107     }
108 
109     /** Subtractive constructor.
110      * <p>Build a relative PVCoordinates from a start and an end position.</p>
111      * <p>The PVCoordinates built will be end - start.</p>
112      * @param start Starting PVCoordinates
113      * @param end ending PVCoordinates
114      */
115     public PVCoordinates(final PVCoordinates start, final PVCoordinates end) {
116         this.position     = end.position.subtract(start.position);
117         this.velocity     = end.velocity.subtract(start.velocity);
118         this.acceleration = end.acceleration.subtract(start.acceleration);
119     }
120 
121     /** Linear constructor.
122      * <p>Build a PVCoordinates from two other ones and corresponding scale factors.</p>
123      * <p>The PVCoordinates built will be a1 * u1 + a2 * u2</p>
124      * @param a1 first scale factor
125      * @param pv1 first base (unscaled) PVCoordinates
126      * @param a2 second scale factor
127      * @param pv2 second base (unscaled) PVCoordinates
128      */
129     public PVCoordinates(final double a1, final PVCoordinates pv1,
130                          final double a2, final PVCoordinates pv2) {
131         position     = new Vector3D(a1, pv1.position,     a2, pv2.position);
132         velocity     = new Vector3D(a1, pv1.velocity,     a2, pv2.velocity);
133         acceleration = new Vector3D(a1, pv1.acceleration, a2, pv2.acceleration);
134     }
135 
136     /** Linear constructor.
137      * <p>Build a PVCoordinates from three other ones and corresponding scale factors.</p>
138      * <p>The PVCoordinates built will be a1 * u1 + a2 * u2 + a3 * u3</p>
139      * @param a1 first scale factor
140      * @param pv1 first base (unscaled) PVCoordinates
141      * @param a2 second scale factor
142      * @param pv2 second base (unscaled) PVCoordinates
143      * @param a3 third scale factor
144      * @param pv3 third base (unscaled) PVCoordinates
145      */
146     public PVCoordinates(final double a1, final PVCoordinates pv1,
147                          final double a2, final PVCoordinates pv2,
148                          final double a3, final PVCoordinates pv3) {
149         position     = new Vector3D(a1, pv1.position,     a2, pv2.position,     a3, pv3.position);
150         velocity     = new Vector3D(a1, pv1.velocity,     a2, pv2.velocity,     a3, pv3.velocity);
151         acceleration = new Vector3D(a1, pv1.acceleration, a2, pv2.acceleration, a3, pv3.acceleration);
152     }
153 
154     /** Linear constructor.
155      * <p>Build a PVCoordinates from four other ones and corresponding scale factors.</p>
156      * <p>The PVCoordinates built will be a1 * u1 + a2 * u2 + a3 * u3 + a4 * u4</p>
157      * @param a1 first scale factor
158      * @param pv1 first base (unscaled) PVCoordinates
159      * @param a2 second scale factor
160      * @param pv2 second base (unscaled) PVCoordinates
161      * @param a3 third scale factor
162      * @param pv3 third base (unscaled) PVCoordinates
163      * @param a4 fourth scale factor
164      * @param pv4 fourth base (unscaled) PVCoordinates
165      */
166     public PVCoordinates(final double a1, final PVCoordinates pv1,
167                          final double a2, final PVCoordinates pv2,
168                          final double a3, final PVCoordinates pv3,
169                          final double a4, final PVCoordinates pv4) {
170         position     = new Vector3D(a1, pv1.position,     a2, pv2.position,
171                                     a3, pv3.position,     a4, pv4.position);
172         velocity     = new Vector3D(a1, pv1.velocity,     a2, pv2.velocity,
173                                     a3, pv3.velocity,     a4, pv4.velocity);
174         acceleration = new Vector3D(a1, pv1.acceleration, a2, pv2.acceleration,
175                                     a3, pv3.acceleration, a4, pv4.acceleration);
176     }
177 
178     /** Builds a PVCoordinates triplet from  a {@link FieldVector3D}&lt;{@link Derivative}&gt;.
179      * <p>
180      * The vector components must have time as their only derivation parameter and
181      * have consistent derivation orders.
182      * </p>
183      * @param p vector with time-derivatives embedded within the coordinates
184      * @param <U> type of the derivative
185      */
186     public <U extends Derivative<U>> PVCoordinates(final FieldVector3D<U> p) {
187         position = new Vector3D(p.getX().getReal(), p.getY().getReal(), p.getZ().getReal());
188         if (p.getX().getOrder() >= 1) {
189             velocity = new Vector3D(p.getX().getPartialDerivative(1),
190                                     p.getY().getPartialDerivative(1),
191                                     p.getZ().getPartialDerivative(1));
192             if (p.getX().getOrder() >= 2) {
193                 acceleration = new Vector3D(p.getX().getPartialDerivative(2),
194                                             p.getY().getPartialDerivative(2),
195                                             p.getZ().getPartialDerivative(2));
196             } else {
197                 acceleration = Vector3D.ZERO;
198             }
199         } else {
200             velocity     = Vector3D.ZERO;
201             acceleration = Vector3D.ZERO;
202         }
203     }
204 
205     /**
206      * Builds PV coordinates with the givne position, zero velocity, and zero
207      * acceleration.
208      *
209      * @param position position vector (m)
210      */
211     public PVCoordinates(final Vector3D position) {
212         this(position, Vector3D.ZERO);
213     }
214 
215     /** Transform the instance to a {@link FieldVector3D}&lt;{@link DerivativeStructure}&gt;.
216      * <p>
217      * The {@link DerivativeStructure} coordinates correspond to time-derivatives up
218      * to the user-specified order.
219      * </p>
220      * @param order derivation order for the vector components (must be either 0, 1 or 2)
221      * @return vector with time-derivatives embedded within the coordinates
222      */
223     public FieldVector3D<DerivativeStructure> toDerivativeStructureVector(final int order) {
224 
225         final DSFactory factory;
226         final DerivativeStructure x;
227         final DerivativeStructure y;
228         final DerivativeStructure z;
229         switch (order) {
230             case 0 :
231                 factory = new DSFactory(1, order);
232                 x = factory.build(position.getX());
233                 y = factory.build(position.getY());
234                 z = factory.build(position.getZ());
235                 break;
236             case 1 :
237                 factory = new DSFactory(1, order);
238                 x = factory.build(position.getX(), velocity.getX());
239                 y = factory.build(position.getY(), velocity.getY());
240                 z = factory.build(position.getZ(), velocity.getZ());
241                 break;
242             case 2 :
243                 factory = new DSFactory(1, order);
244                 x = factory.build(position.getX(), velocity.getX(), acceleration.getX());
245                 y = factory.build(position.getY(), velocity.getY(), acceleration.getY());
246                 z = factory.build(position.getZ(), velocity.getZ(), acceleration.getZ());
247                 break;
248             default :
249                 throw new OrekitException(OrekitMessages.OUT_OF_RANGE_DERIVATION_ORDER, order);
250         }
251 
252         return new FieldVector3D<>(x, y, z);
253 
254     }
255 
256     /** Transform the instance to a {@link FieldVector3D}&lt;{@link UnivariateDerivative1}&gt;.
257      * <p>
258      * The {@link UnivariateDerivative1} coordinates correspond to time-derivatives up
259      * to the order 1.
260      * </p>
261      * @return vector with time-derivatives embedded within the coordinates
262      * @see #toUnivariateDerivative2Vector()
263      * @since 10.2
264      */
265     public FieldVector3D<UnivariateDerivative1> toUnivariateDerivative1Vector() {
266 
267         final UnivariateDerivative1 x = new UnivariateDerivative1(position.getX(), velocity.getX());
268         final UnivariateDerivative1 y = new UnivariateDerivative1(position.getY(), velocity.getY());
269         final UnivariateDerivative1 z = new UnivariateDerivative1(position.getZ(), velocity.getZ());
270 
271         return new FieldVector3D<>(x, y, z);
272     }
273 
274     /** Transform the instance to a {@link FieldVector3D}&lt;{@link UnivariateDerivative2}&gt;.
275      * <p>
276      * The {@link UnivariateDerivative2} coordinates correspond to time-derivatives up
277      * to the order 2.
278      * </p>
279      * @return vector with time-derivatives embedded within the coordinates
280      * @see #toUnivariateDerivative1Vector()
281      * @since 10.2
282      */
283     public FieldVector3D<UnivariateDerivative2> toUnivariateDerivative2Vector() {
284 
285         final UnivariateDerivative2 x = new UnivariateDerivative2(position.getX(), velocity.getX(), acceleration.getX());
286         final UnivariateDerivative2 y = new UnivariateDerivative2(position.getY(), velocity.getY(), acceleration.getY());
287         final UnivariateDerivative2 z = new UnivariateDerivative2(position.getZ(), velocity.getZ(), acceleration.getZ());
288 
289         return new FieldVector3D<>(x, y, z);
290     }
291 
292     /** Transform the instance to a {@link FieldPVCoordinates}&lt;{@link DerivativeStructure}&gt;.
293      * <p>
294      * The {@link DerivativeStructure} coordinates correspond to time-derivatives up
295      * to the user-specified order. As both the instance components {@link #getPosition() position},
296      * {@link #getVelocity() velocity} and {@link #getAcceleration() acceleration} and the
297      * {@link DerivativeStructure#getPartialDerivative(int...) derivatives} of the components
298      * holds time-derivatives, there are several ways to retrieve these derivatives. If for example
299      * the {@code order} is set to 2, then both {@code pv.getPosition().getX().getPartialDerivative(2)},
300      * {@code pv.getVelocity().getX().getPartialDerivative(1)} and
301      * {@code pv.getAcceleration().getX().getValue()} return the exact same value.
302      * </p>
303      * <p>
304      * If derivation order is 1, the first derivative of acceleration will be computed as a
305      * Keplerian-only jerk. If derivation order is 2, the second derivative of velocity (which
306      * is also the first derivative of acceleration) will be computed as a Keplerian-only jerk,
307      * and the second derivative of acceleration will be computed as a Keplerian-only jounce.
308      * </p>
309      * @param order derivation order for the vector components (must be either 0, 1 or 2)
310      * @return pv coordinates with time-derivatives embedded within the coordinates
311      * @since 9.2
312      */
313     public FieldPVCoordinates<DerivativeStructure> toDerivativeStructurePV(final int order) {
314 
315         final DSFactory factory;
316         final DerivativeStructure x0;
317         final DerivativeStructure y0;
318         final DerivativeStructure z0;
319         final DerivativeStructure x1;
320         final DerivativeStructure y1;
321         final DerivativeStructure z1;
322         final DerivativeStructure x2;
323         final DerivativeStructure y2;
324         final DerivativeStructure z2;
325         switch (order) {
326             case 0 :
327                 factory = new DSFactory(1, order);
328                 x0 = factory.build(position.getX());
329                 y0 = factory.build(position.getY());
330                 z0 = factory.build(position.getZ());
331                 x1 = factory.build(velocity.getX());
332                 y1 = factory.build(velocity.getY());
333                 z1 = factory.build(velocity.getZ());
334                 x2 = factory.build(acceleration.getX());
335                 y2 = factory.build(acceleration.getY());
336                 z2 = factory.build(acceleration.getZ());
337                 break;
338             case 1 : {
339                 factory = new DSFactory(1, order);
340                 final double   r2            = position.getNormSq();
341                 final double   r             = FastMath.sqrt(r2);
342                 final double   pvOr2         = Vector3D.dotProduct(position, velocity) / r2;
343                 final double   a             = acceleration.getNorm();
344                 final double   aOr           = a / r;
345                 final Vector3D keplerianJerk = new Vector3D(-3 * pvOr2, acceleration, -aOr, velocity);
346                 x0 = factory.build(position.getX(),     velocity.getX());
347                 y0 = factory.build(position.getY(),     velocity.getY());
348                 z0 = factory.build(position.getZ(),     velocity.getZ());
349                 x1 = factory.build(velocity.getX(),     acceleration.getX());
350                 y1 = factory.build(velocity.getY(),     acceleration.getY());
351                 z1 = factory.build(velocity.getZ(),     acceleration.getZ());
352                 x2 = factory.build(acceleration.getX(), keplerianJerk.getX());
353                 y2 = factory.build(acceleration.getY(), keplerianJerk.getY());
354                 z2 = factory.build(acceleration.getZ(), keplerianJerk.getZ());
355                 break;
356             }
357             case 2 : {
358                 factory = new DSFactory(1, order);
359                 final double   r2              = position.getNormSq();
360                 final double   r               = FastMath.sqrt(r2);
361                 final double   pvOr2           = Vector3D.dotProduct(position, velocity) / r2;
362                 final double   a               = acceleration.getNorm();
363                 final double   aOr             = a / r;
364                 final Vector3D keplerianJerk   = new Vector3D(-3 * pvOr2, acceleration, -aOr, velocity);
365                 final double   v2              = velocity.getNormSq();
366                 final double   pa              = Vector3D.dotProduct(position, acceleration);
367                 final double   aj              = Vector3D.dotProduct(acceleration, keplerianJerk);
368                 final Vector3D keplerianJounce = new Vector3D(-3 * (v2 + pa) / r2 + 15 * pvOr2 * pvOr2 - aOr, acceleration,
369                                                               4 * aOr * pvOr2 - aj / (a * r), velocity);
370                 x0 = factory.build(position.getX(),     velocity.getX(),      acceleration.getX());
371                 y0 = factory.build(position.getY(),     velocity.getY(),      acceleration.getY());
372                 z0 = factory.build(position.getZ(),     velocity.getZ(),      acceleration.getZ());
373                 x1 = factory.build(velocity.getX(),     acceleration.getX(),  keplerianJerk.getX());
374                 y1 = factory.build(velocity.getY(),     acceleration.getY(),  keplerianJerk.getY());
375                 z1 = factory.build(velocity.getZ(),     acceleration.getZ(),  keplerianJerk.getZ());
376                 x2 = factory.build(acceleration.getX(), keplerianJerk.getX(), keplerianJounce.getX());
377                 y2 = factory.build(acceleration.getY(), keplerianJerk.getY(), keplerianJounce.getY());
378                 z2 = factory.build(acceleration.getZ(), keplerianJerk.getZ(), keplerianJounce.getZ());
379                 break;
380             }
381             default :
382                 throw new OrekitException(OrekitMessages.OUT_OF_RANGE_DERIVATION_ORDER, order);
383         }
384 
385         return new FieldPVCoordinates<>(new FieldVector3D<>(x0, y0, z0),
386                                         new FieldVector3D<>(x1, y1, z1),
387                                         new FieldVector3D<>(x2, y2, z2));
388 
389     }
390 
391     /** Transform the instance to a {@link FieldPVCoordinates}&lt;{@link UnivariateDerivative1}&gt;.
392      * <p>
393      * The {@link UnivariateDerivative1} coordinates correspond to time-derivatives up
394      * to the order 1.
395      * The first derivative of acceleration will be computed as a Keplerian-only jerk.
396      * </p>
397      * @return pv coordinates with time-derivatives embedded within the coordinates
398      * @since 10.2
399      */
400     public FieldPVCoordinates<UnivariateDerivative1> toUnivariateDerivative1PV() {
401 
402         final double   r2            = position.getNormSq();
403         final double   r             = FastMath.sqrt(r2);
404         final double   pvOr2         = Vector3D.dotProduct(position, velocity) / r2;
405         final double   a             = acceleration.getNorm();
406         final double   aOr           = a / r;
407         final Vector3D keplerianJerk = new Vector3D(-3 * pvOr2, acceleration, -aOr, velocity);
408 
409         final UnivariateDerivative1 x0 = new UnivariateDerivative1(position.getX(),     velocity.getX());
410         final UnivariateDerivative1 y0 = new UnivariateDerivative1(position.getY(),     velocity.getY());
411         final UnivariateDerivative1 z0 = new UnivariateDerivative1(position.getZ(),     velocity.getZ());
412         final UnivariateDerivative1 x1 = new UnivariateDerivative1(velocity.getX(),     acceleration.getX());
413         final UnivariateDerivative1 y1 = new UnivariateDerivative1(velocity.getY(),     acceleration.getY());
414         final UnivariateDerivative1 z1 = new UnivariateDerivative1(velocity.getZ(),     acceleration.getZ());
415         final UnivariateDerivative1 x2 = new UnivariateDerivative1(acceleration.getX(), keplerianJerk.getX());
416         final UnivariateDerivative1 y2 = new UnivariateDerivative1(acceleration.getY(), keplerianJerk.getY());
417         final UnivariateDerivative1 z2 = new UnivariateDerivative1(acceleration.getZ(), keplerianJerk.getZ());
418 
419         return new FieldPVCoordinates<>(new FieldVector3D<>(x0, y0, z0),
420                                         new FieldVector3D<>(x1, y1, z1),
421                                         new FieldVector3D<>(x2, y2, z2));
422 
423     }
424 
425     /** Transform the instance to a {@link FieldPVCoordinates}&lt;{@link UnivariateDerivative2}&gt;.
426      * <p>
427      * The {@link UnivariateDerivative2} coordinates correspond to time-derivatives up
428      * to the order 2.
429      * As derivation order is 2, the second derivative of velocity (which
430      * is also the first derivative of acceleration) will be computed as a Keplerian-only jerk,
431      * and the second derivative of acceleration will be computed as a Keplerian-only jounce.
432      * </p>
433      * @return pv coordinates with time-derivatives embedded within the coordinates
434      * @since 10.2
435      */
436     public FieldPVCoordinates<UnivariateDerivative2> toUnivariateDerivative2PV() {
437 
438         final double   r2              = position.getNormSq();
439         final double   r               = FastMath.sqrt(r2);
440         final double   pvOr2           = Vector3D.dotProduct(position, velocity) / r2;
441         final double   a               = acceleration.getNorm();
442         final double   aOr             = a / r;
443         final Vector3D keplerianJerk   = new Vector3D(-3 * pvOr2, acceleration, -aOr, velocity);
444         final double   v2              = velocity.getNormSq();
445         final double   pa              = Vector3D.dotProduct(position, acceleration);
446         final double   aj              = Vector3D.dotProduct(acceleration, keplerianJerk);
447         final Vector3D keplerianJounce = new Vector3D(-3 * (v2 + pa) / r2 + 15 * pvOr2 * pvOr2 - aOr, acceleration,
448                                                       4 * aOr * pvOr2 - aj / (a * r), velocity);
449 
450         final UnivariateDerivative2 x0 = new UnivariateDerivative2(position.getX(),     velocity.getX(),      acceleration.getX());
451         final UnivariateDerivative2 y0 = new UnivariateDerivative2(position.getY(),     velocity.getY(),      acceleration.getY());
452         final UnivariateDerivative2 z0 = new UnivariateDerivative2(position.getZ(),     velocity.getZ(),      acceleration.getZ());
453         final UnivariateDerivative2 x1 = new UnivariateDerivative2(velocity.getX(),     acceleration.getX(),  keplerianJerk.getX());
454         final UnivariateDerivative2 y1 = new UnivariateDerivative2(velocity.getY(),     acceleration.getY(),  keplerianJerk.getY());
455         final UnivariateDerivative2 z1 = new UnivariateDerivative2(velocity.getZ(),     acceleration.getZ(),  keplerianJerk.getZ());
456         final UnivariateDerivative2 x2 = new UnivariateDerivative2(acceleration.getX(), keplerianJerk.getX(), keplerianJounce.getX());
457         final UnivariateDerivative2 y2 = new UnivariateDerivative2(acceleration.getY(), keplerianJerk.getY(), keplerianJounce.getY());
458         final UnivariateDerivative2 z2 = new UnivariateDerivative2(acceleration.getZ(), keplerianJerk.getZ(), keplerianJounce.getZ());
459 
460         return new FieldPVCoordinates<>(new FieldVector3D<>(x0, y0, z0),
461                                         new FieldVector3D<>(x1, y1, z1),
462                                         new FieldVector3D<>(x2, y2, z2));
463 
464     }
465 
466     /** Estimate velocity between two positions.
467      * <p>Estimation is based on a simple fixed velocity translation
468      * during the time interval between the two positions.</p>
469      * @param start start position
470      * @param end end position
471      * @param dt time elapsed between the dates of the two positions
472      * @return velocity allowing to go from start to end positions
473      */
474     public static Vector3D estimateVelocity(final Vector3D start, final Vector3D end, final double dt) {
475         final double scale = 1.0 / dt;
476         return new Vector3D(scale, end, -scale, start);
477     }
478 
479     /** Get a time-shifted state.
480      * <p>
481      * The state can be slightly shifted to close dates. This shift is based on
482      * a simple Taylor expansion. It is <em>not</em> intended as a replacement for
483      * proper orbit propagation (it is not even Keplerian!) but should be sufficient
484      * for either small time shifts or coarse accuracy.
485      * </p>
486      * @param dt time shift in seconds
487      * @return a new state, shifted with respect to the instance (which is immutable)
488      */
489     public PVCoordinates shiftedBy(final double dt) {
490         return new PVCoordinates(positionShiftedBy(dt),
491                                  new Vector3D(1, velocity, dt, acceleration),
492                                  acceleration);
493     }
494 
495     /**
496      * Get a time-shifted position. Same as {@link #shiftedBy(double)} except
497      * that only the sifted position is returned.
498      * <p>
499      * The state can be slightly shifted to close dates. This shift is based on
500      * a simple Taylor expansion. It is <em>not</em> intended as a replacement
501      * for proper orbit propagation (it is not even Keplerian!) but should be
502      * sufficient for either small time shifts or coarse accuracy.
503      * </p>
504      *
505      * @param dt time shift in seconds
506      * @return a new state, shifted with respect to the instance (which is
507      * immutable)
508      */
509     public Vector3D positionShiftedBy(final double dt) {
510         return new Vector3D(1, position, dt, velocity, 0.5 * dt * dt, acceleration);
511     }
512 
513     /** Gets the position.
514      * @return the position vector (m).
515      */
516     public Vector3D getPosition() {
517         return position;
518     }
519 
520     /** Gets the velocity.
521      * @return the velocity vector (m/s).
522      */
523     public Vector3D getVelocity() {
524         return velocity;
525     }
526 
527     /** Gets the acceleration.
528      * @return the acceleration vector (m/s²).
529      */
530     public Vector3D getAcceleration() {
531         return acceleration;
532     }
533 
534     /** Gets the momentum.
535      * <p>This vector is the p &otimes; v where p is position, v is velocity
536      * and &otimes; is cross product. To get the real physical angular momentum
537      * you need to multiply this vector by the mass.</p>
538      * <p>The returned vector is recomputed each time this method is called, it
539      * is not cached.</p>
540      * @return a new instance of the momentum vector (m²/s).
541      */
542     public Vector3D getMomentum() {
543         return Vector3D.crossProduct(position, velocity);
544     }
545 
546     /**
547      * Get the angular velocity (spin) of this point as seen from the origin.
548      *
549      * <p> The angular velocity vector is parallel to the {@link #getMomentum()
550      * angular momentum} and is computed by ω = p &times; v / ||p||²
551      *
552      * @return the angular velocity vector
553      * @see <a href="http://en.wikipedia.org/wiki/Angular_velocity">Angular Velocity on
554      *      Wikipedia</a>
555      */
556     public Vector3D getAngularVelocity() {
557         return this.getMomentum().scalarMultiply(1.0 / this.getPosition().getNormSq());
558     }
559 
560     /** Get the opposite of the instance.
561      * @return a new position-velocity which is opposite to the instance
562      */
563     public PVCoordinates negate() {
564         return new PVCoordinates(position.negate(), velocity.negate(), acceleration.negate());
565     }
566 
567     /** Normalize the position part of the instance.
568      * <p>
569      * The computed coordinates first component (position) will be a
570      * normalized vector, the second component (velocity) will be the
571      * derivative of the first component (hence it will generally not
572      * be normalized), and the third component (acceleration) will be the
573      * derivative of the second component (hence it will generally not
574      * be normalized).
575      * </p>
576      * @return a new instance, with first component normalized and
577      * remaining component computed to have consistent derivatives
578      */
579     public PVCoordinates normalize() {
580         final double   inv     = 1.0 / position.getNorm();
581         final Vector3D u       = new Vector3D(inv, position);
582         final Vector3D v       = new Vector3D(inv, velocity);
583         final Vector3D w       = new Vector3D(inv, acceleration);
584         final double   uv      = Vector3D.dotProduct(u, v);
585         final double   v2      = Vector3D.dotProduct(v, v);
586         final double   uw      = Vector3D.dotProduct(u, w);
587         final Vector3D uDot    = new Vector3D(1, v, -uv, u);
588         final Vector3D uDotDot = new Vector3D(1, w, -2 * uv, v, 3 * uv * uv - v2 - uw, u);
589         return new PVCoordinates(u, uDot, uDotDot);
590     }
591 
592     /** Compute the cross-product of two instances.
593      * @param pv1 first instances
594      * @param pv2 second instances
595      * @return the cross product v1 ^ v2 as a new instance
596      */
597     public static PVCoordinates crossProduct(final PVCoordinates pv1, final PVCoordinates pv2) {
598         final Vector3D p1 = pv1.position;
599         final Vector3D v1 = pv1.velocity;
600         final Vector3D a1 = pv1.acceleration;
601         final Vector3D p2 = pv2.position;
602         final Vector3D v2 = pv2.velocity;
603         final Vector3D a2 = pv2.acceleration;
604         return new PVCoordinates(Vector3D.crossProduct(p1, p2),
605                                  new Vector3D(1, Vector3D.crossProduct(p1, v2),
606                                               1, Vector3D.crossProduct(v1, p2)),
607                                  new Vector3D(1, Vector3D.crossProduct(p1, a2),
608                                               2, Vector3D.crossProduct(v1, v2),
609                                               1, Vector3D.crossProduct(a1, p2)));
610     }
611 
612     /** Return a string representation of this position/velocity pair.
613      * @return string representation of this position/velocity pair
614      */
615     public String toString() {
616         final String comma = ", ";
617         return new StringBuilder().append('{').append("P(").
618                 append(position.getX()).append(comma).
619                 append(position.getY()).append(comma).
620                 append(position.getZ()).append("), V(").
621                 append(velocity.getX()).append(comma).
622                 append(velocity.getY()).append(comma).
623                 append(velocity.getZ()).append("), A(").
624                 append(acceleration.getX()).append(comma).
625                 append(acceleration.getY()).append(comma).
626                 append(acceleration.getZ()).append(")}").toString();
627     }
628 
629     /** Replace the instance with a data transfer object for serialization.
630      * @return data transfer object that will be serialized
631      */
632     private Object writeReplace() {
633         return new DTO(this);
634     }
635 
636     /** {@inheritDoc} */
637     @Override
638     public PVCoordinates blendArithmeticallyWith(final PVCoordinates other, final double blendingValue)
639             throws MathIllegalArgumentException {
640         final Vector3D blendedPosition     = position.blendArithmeticallyWith(other.position, blendingValue);
641         final Vector3D blendedVelocity     = velocity.blendArithmeticallyWith(other.velocity, blendingValue);
642         final Vector3D blendedAcceleration = acceleration.blendArithmeticallyWith(other.acceleration, blendingValue);
643 
644         return new PVCoordinates(blendedPosition, blendedVelocity, blendedAcceleration);
645     }
646 
647     /** Internal class used only for serialization. */
648     private static class DTO implements Serializable {
649 
650         /** Serializable UID. */
651         private static final long serialVersionUID = 20140723L;
652 
653         /** Double values. */
654         private final double[] d;
655 
656         /** Simple constructor.
657          * @param pv instance to serialize
658          */
659         private DTO(final PVCoordinates pv) {
660             this.d = new double[] {
661                 pv.getPosition().getX(),     pv.getPosition().getY(),     pv.getPosition().getZ(),
662                 pv.getVelocity().getX(),     pv.getVelocity().getY(),     pv.getVelocity().getZ(),
663                 pv.getAcceleration().getX(), pv.getAcceleration().getY(), pv.getAcceleration().getZ(),
664             };
665         }
666 
667         /** Replace the deserialized data transfer object with a {@link PVCoordinates}.
668          * @return replacement {@link PVCoordinates}
669          */
670         private Object readResolve() {
671             return new PVCoordinates(new Vector3D(d[0], d[1], d[2]),
672                                      new Vector3D(d[3], d[4], d[5]),
673                                      new Vector3D(d[6], d[7], d[8]));
674         }
675 
676     }
677 
678 }