CA2814882C - Method and apparatus for determining the relative position between two receivers and use of the apparatus for stabilizing suspended loads - Google Patents
Method and apparatus for determining the relative position between two receivers and use of the apparatus for stabilizing suspended loads Download PDFInfo
- Publication number
- CA2814882C CA2814882C CA2814882A CA2814882A CA2814882C CA 2814882 C CA2814882 C CA 2814882C CA 2814882 A CA2814882 A CA 2814882A CA 2814882 A CA2814882 A CA 2814882A CA 2814882 C CA2814882 C CA 2814882C
- Authority
- CA
- Canada
- Prior art keywords
- distance vector
- phase ambiguities
- phase
- integer
- receivers
- Prior art date
- Legal status (The legal status is an assumption and is not a legal conclusion. Google has not performed a legal analysis and makes no representation as to the accuracy of the status listed.)
- Expired - Fee Related
Links
Classifications
-
- G—PHYSICS
- G01—MEASURING; TESTING
- G01S—RADIO DIRECTION-FINDING; RADIO NAVIGATION; DETERMINING DISTANCE OR VELOCITY BY USE OF RADIO WAVES; LOCATING OR PRESENCE-DETECTING BY USE OF THE REFLECTION OR RERADIATION OF RADIO WAVES; ANALOGOUS ARRANGEMENTS USING OTHER WAVES
- G01S19/00—Satellite radio beacon positioning systems; Determining position, velocity or attitude using signals transmitted by such systems
- G01S19/38—Determining a navigation solution using signals transmitted by a satellite radio beacon positioning system
- G01S19/39—Determining a navigation solution using signals transmitted by a satellite radio beacon positioning system the satellite radio beacon positioning system transmitting time-stamped messages, e.g. GPS [Global Positioning System], GLONASS [Global Orbiting Navigation Satellite System] or GALILEO
- G01S19/42—Determining position
- G01S19/51—Relative positioning
-
- G—PHYSICS
- G01—MEASURING; TESTING
- G01S—RADIO DIRECTION-FINDING; RADIO NAVIGATION; DETERMINING DISTANCE OR VELOCITY BY USE OF RADIO WAVES; LOCATING OR PRESENCE-DETECTING BY USE OF THE REFLECTION OR RERADIATION OF RADIO WAVES; ANALOGOUS ARRANGEMENTS USING OTHER WAVES
- G01S19/00—Satellite radio beacon positioning systems; Determining position, velocity or attitude using signals transmitted by such systems
- G01S19/38—Determining a navigation solution using signals transmitted by a satellite radio beacon positioning system
- G01S19/39—Determining a navigation solution using signals transmitted by a satellite radio beacon positioning system the satellite radio beacon positioning system transmitting time-stamped messages, e.g. GPS [Global Positioning System], GLONASS [Global Orbiting Navigation Satellite System] or GALILEO
- G01S19/42—Determining position
- G01S19/43—Determining position using carrier phase measurements, e.g. kinematic positioning; using long or short baseline interferometry
- G01S19/44—Carrier phase ambiguity resolution; Floating ambiguity; LAMBDA [Least-squares AMBiguity Decorrelation Adjustment] method
Landscapes
- Engineering & Computer Science (AREA)
- Radar, Positioning & Navigation (AREA)
- Remote Sensing (AREA)
- Computer Networks & Wireless Communication (AREA)
- Physics & Mathematics (AREA)
- General Physics & Mathematics (AREA)
- Position Fixing By Use Of Radio Waves (AREA)
Abstract
The invention proposes a method and an apparatus for determining the relative positions between two receivers (10, 11) for satellite navigation. In the case of the method and the apparatus, phase measurements are carried out on carrier signals (3) of a satellite navigation system (1) by the receivers (10, 11). An evaluation unit (14) determines the relative position of the receivers (10, 11) using the code and phase measurements by determining the integer phase ambiguities and the distance vector (15) which describes the relative position. For this purpose, the evaluation unit (14) optimizes an evaluation function which, in addition to a first measure for evaluating the integer phase ambiguities, comprises a second measure which is used to determine the difference between the distance vector (15), which describes the relative position and is assigned to the integer phase ambiguities, and a distance vector of a predetermined length.
Description
METHOD AND APPARATUS FOR DETERMINING THE RELATIVE POSITION
BETWEEN TWO RECEIVERS AND USE OF THE APPARATUS FOR
STABILIZING SUSPENDED LOADS
Description:
The invention relates to a method for determining the relative position between two receivers for satellite navigation, in which phase measurements on carrier signals of the satellite navigation system are performed by the receivers and in which the relative position of the receivers is determined by an evaluation unit using code and phase measurements.
The invention further relates to an apparatus for determining the relative position between two receivers as well as the use of the apparatus for stabilizing suspended loads.
A method for determining the relative position between two receivers is known from the publication GIORGI G., TEUNISSEN P., VERHAGEN S.: Reducing the Time-To-Fix for Stand-Alone Single Frequency GNSS Attitude Determination, Proc. of Intern. Techn. Meeting of the Inst. of Navigation (ION-ITM), San Diego, January 2010. Determining a distance vector describing the relative position between two receivers basically comes down to solving a minimization problem in the framework of a least square method. The solution for the distance vector shall be found under the constraint that the length of the distance vector has a predetermined value. In the known method, the distance vector is presented in polar coordinates. This, however, results in strong non-linearities with respect to the angles of the polar coordinates, that are overcome by separating the distance vector in an initial estimate and in a variation, which scales linearly with the angles.
BETWEEN TWO RECEIVERS AND USE OF THE APPARATUS FOR
STABILIZING SUSPENDED LOADS
Description:
The invention relates to a method for determining the relative position between two receivers for satellite navigation, in which phase measurements on carrier signals of the satellite navigation system are performed by the receivers and in which the relative position of the receivers is determined by an evaluation unit using code and phase measurements.
The invention further relates to an apparatus for determining the relative position between two receivers as well as the use of the apparatus for stabilizing suspended loads.
A method for determining the relative position between two receivers is known from the publication GIORGI G., TEUNISSEN P., VERHAGEN S.: Reducing the Time-To-Fix for Stand-Alone Single Frequency GNSS Attitude Determination, Proc. of Intern. Techn. Meeting of the Inst. of Navigation (ION-ITM), San Diego, January 2010. Determining a distance vector describing the relative position between two receivers basically comes down to solving a minimization problem in the framework of a least square method. The solution for the distance vector shall be found under the constraint that the length of the distance vector has a predetermined value. In the known method, the distance vector is presented in polar coordinates. This, however, results in strong non-linearities with respect to the angles of the polar coordinates, that are overcome by separating the distance vector in an initial estimate and in a variation, which scales linearly with the angles.
2 Thus, the system of equations, which is non-linear with respect to the angles of the polar coordinates, can be linearized and be solved by conventional methods.
One drawback of the known method is that the linearization can only be performed if the initial distance vector can be determined with sufficient accuracy. That is the case, if the length of the distance vector is significantly bigger than the error on the initial absolute determination of the positions of both receivers. If the distance vector has a length greater 100 meters, that is usually the case.
TEUNISSEN, P.: Least-squares estimation of the integer ambiguities, Invited lecture, Section IV, Theory and Methodology, IAG General Meeting, Beijing, China, 1993 describes various methods for resolving phase ambiguities.
TEUNISSEN, P.: The least-squares ambiguity decorrelation adjustment: a method for fast GPS ambiguity estimation, J.
of Geodesy, volume 70, pages 65-82, 1995 describes a decorrelation method for resolving phase ambiguities.
HENKEL, P.: Bootstrapping with Multi-Frequency Mixed Code Carrier Linear Combinations and Partial Integer Decorrelation in the Presence of Biases, Proc. of the IAG
Scient. Assembly, Buenos Aires, Argentina, September 2009 studies code and phase combinations for improving the resolution of phase ambiguities.
From EP 1 972 959 Al, various methods for resolving phase ambiguities of linear combinations of carrier signals of a satellite navigation system are known.
From WO 2009/125011, a method is further known for tracking phases of a plurality of carrier signals.
One drawback of the known method is that the linearization can only be performed if the initial distance vector can be determined with sufficient accuracy. That is the case, if the length of the distance vector is significantly bigger than the error on the initial absolute determination of the positions of both receivers. If the distance vector has a length greater 100 meters, that is usually the case.
TEUNISSEN, P.: Least-squares estimation of the integer ambiguities, Invited lecture, Section IV, Theory and Methodology, IAG General Meeting, Beijing, China, 1993 describes various methods for resolving phase ambiguities.
TEUNISSEN, P.: The least-squares ambiguity decorrelation adjustment: a method for fast GPS ambiguity estimation, J.
of Geodesy, volume 70, pages 65-82, 1995 describes a decorrelation method for resolving phase ambiguities.
HENKEL, P.: Bootstrapping with Multi-Frequency Mixed Code Carrier Linear Combinations and Partial Integer Decorrelation in the Presence of Biases, Proc. of the IAG
Scient. Assembly, Buenos Aires, Argentina, September 2009 studies code and phase combinations for improving the resolution of phase ambiguities.
From EP 1 972 959 Al, various methods for resolving phase ambiguities of linear combinations of carrier signals of a satellite navigation system are known.
From WO 2009/125011, a method is further known for tracking phases of a plurality of carrier signals.
3 Proceeding from this related prior art, the invention is based on the object to provide an improved method for determining the relative position between two receivers of a satellite navigation system.
This object is achieved by an apparatus having the features of the independent claim. Advantageous embodiments and refinements are specified in claims dependent thereon.
In the method, the evaluation unit determines the integer phase ambiguities and the distance vector describing the relative position by minimizing an evaluation function, which, in addition to a first measure for evaluating the consistency of the integer phase ambiguities with the phase measurements, comprises a second measure, which determines the deviation of the distance vector, which describes the relative position and is associated with the integer phase ambiguities and whose length is not predetermined, from a distance vector of a predetermined length. The method thus determines the one distance vector, which, on the one hand, corresponds as well as possible with known previous knowledge, and which, on the other hand, is associated with phase ambiguities, which are as much as possible consistent with phase measurements. By such a method, the relative position of both receivers can also be determined, if the error on the determination of the absolute position of the receivers is not negligible in view of the distance between the receivers.
In one embodiment of the method, the real-valued phase ambiguities are determined first and then the integer phase ambiguities and the distance vector describing the relative position, are determined by optimizing the evaluation function. The first measure of the evaluation function includes a norm, which can be used for determining the variation of the real-valued phase ambiguities from the
This object is achieved by an apparatus having the features of the independent claim. Advantageous embodiments and refinements are specified in claims dependent thereon.
In the method, the evaluation unit determines the integer phase ambiguities and the distance vector describing the relative position by minimizing an evaluation function, which, in addition to a first measure for evaluating the consistency of the integer phase ambiguities with the phase measurements, comprises a second measure, which determines the deviation of the distance vector, which describes the relative position and is associated with the integer phase ambiguities and whose length is not predetermined, from a distance vector of a predetermined length. The method thus determines the one distance vector, which, on the one hand, corresponds as well as possible with known previous knowledge, and which, on the other hand, is associated with phase ambiguities, which are as much as possible consistent with phase measurements. By such a method, the relative position of both receivers can also be determined, if the error on the determination of the absolute position of the receivers is not negligible in view of the distance between the receivers.
In one embodiment of the method, the real-valued phase ambiguities are determined first and then the integer phase ambiguities and the distance vector describing the relative position, are determined by optimizing the evaluation function. The first measure of the evaluation function includes a norm, which can be used for determining the variation of the real-valued phase ambiguities from the
4 integer phase ambiguities. The second measure further includes a norm, which can be used for determining the deviation of the distance vector, that is associated with the integer phase ambiguities and whose length is not predetermined, from the distance vector of predetermined length. The method is therefore used for searching for the one distance vector, whose associated integer valued phase ambiguities is as close to the real-valued phase ambiguities as possible and that coincides with the predetermined distance vector as precisely as possible.
The method allows to determine the distance vector accurately even without a precise previous knowledge on the direction of the distance vector.
The accuracy in determining the distance vector can be further enhanced, if it is additionally required that the distance vector to be determined must have a previously known length. This may be achieved, for instance, by using a search evaluation function and searching for the one distance vector, that comprises a predetermined length and has the smallest distance to the distance vector, which is associated with the integer phase ambiguities and whose length is not predetermined. In this two-step method, the real-valued phase ambiguities are initially estimated in a first method step, and candidates for the integer phase ambiguities are searched, which are within the error limits of the real-valued phase ambiguities. In a second method step, the distance vector to a selected candidate for the integer phase ambiguities is determined, and a further distance vector of previously known length is searched such, that, using the search evaluation function, the one distance vector is searched, that comprises the predetermined length and the smallest distance to the distance vector, which is associated with the integer phase ambiguities and which has no predetermined length. As soon as the searched distance vector has converged to a particular distance vector, the evaluation function can be evaluated. The second method step is repeated for all candidates of the integer phase ambiguities. Among all candidates for the integer phase ambiguities, the one
The method allows to determine the distance vector accurately even without a precise previous knowledge on the direction of the distance vector.
The accuracy in determining the distance vector can be further enhanced, if it is additionally required that the distance vector to be determined must have a previously known length. This may be achieved, for instance, by using a search evaluation function and searching for the one distance vector, that comprises a predetermined length and has the smallest distance to the distance vector, which is associated with the integer phase ambiguities and whose length is not predetermined. In this two-step method, the real-valued phase ambiguities are initially estimated in a first method step, and candidates for the integer phase ambiguities are searched, which are within the error limits of the real-valued phase ambiguities. In a second method step, the distance vector to a selected candidate for the integer phase ambiguities is determined, and a further distance vector of previously known length is searched such, that, using the search evaluation function, the one distance vector is searched, that comprises the predetermined length and the smallest distance to the distance vector, which is associated with the integer phase ambiguities and which has no predetermined length. As soon as the searched distance vector has converged to a particular distance vector, the evaluation function can be evaluated. The second method step is repeated for all candidates of the integer phase ambiguities. Among all candidates for the integer phase ambiguities, the one
5 vector of the integer phase ambiguities is chosen as an optimum, that minimizes the evaluation function together with the associated distance vector.
The search evaluation function can be a Lagrange function, that enforces a given length of the distance vector associated with the integer phase ambiguities.
In an alternative embodiment of the method, the first measure for evaluating the integer phase ambiguities comprises a norm, by which a deviation of the measured code and phase values from calculated code and phase values is determined. The calculated code and phase values have been calculated by means of the integer phase ambiguities and the distance vector. The measured code and phase values, however, have been detected in the control circuits of the receiver (phase lock loop, delay lock loop) during the code and phase measurements. In the modified embodiment of the method, the second measure of the evaluation function further comprises a deviation of a norm of the distance vector from a predetermined length. The relative position of both receivers can also be determined with high accuracy using this method, even if the errors on the determination of the absolute position of the receivers were not be negligible in view of the distance.
If in the modified embodiment the first measure is connected to the second measure by a coupling parameter, by which the influence of the second measurement on the determination of the integer phase ambiguities and the distance can be adjusted, the predetermined length does not need to be known exactly in advance.
The search evaluation function can be a Lagrange function, that enforces a given length of the distance vector associated with the integer phase ambiguities.
In an alternative embodiment of the method, the first measure for evaluating the integer phase ambiguities comprises a norm, by which a deviation of the measured code and phase values from calculated code and phase values is determined. The calculated code and phase values have been calculated by means of the integer phase ambiguities and the distance vector. The measured code and phase values, however, have been detected in the control circuits of the receiver (phase lock loop, delay lock loop) during the code and phase measurements. In the modified embodiment of the method, the second measure of the evaluation function further comprises a deviation of a norm of the distance vector from a predetermined length. The relative position of both receivers can also be determined with high accuracy using this method, even if the errors on the determination of the absolute position of the receivers were not be negligible in view of the distance.
If in the modified embodiment the first measure is connected to the second measure by a coupling parameter, by which the influence of the second measurement on the determination of the integer phase ambiguities and the distance can be adjusted, the predetermined length does not need to be known exactly in advance.
6 In optimizing the evaluation function, advantageously, various integer phase ambiguities are subsequently applied to the evaluation function. The integer phase ambiguities can then be selected by means of a search tree, whose branches correspond to series of various phase ambiguities and in which search tree branches with a phase ambiguity or a length of the distance vector beyond error limits are excluded.
The precision of the method can further be improved, if linear combinations of the phase measurements are formed and if phase ambiguities thereof are searched.
For further improving the accuracy, the ratio of the wavelength to a standard deviation of the noise of the linear combination is maximized by the geometry preserving linear combination.
Additionally, it is also possible to combine the phase measurements with code measurements for increasing the wavelength of the linear combination of the phase measurements and for thereby enhancing the reliability of the estimates.
The method can be performed using an apparatus, that comprises at least two receivers, which perform code and phase measurements on navigation signals, and that is provided with an evaluation unit, by which a distance vector between the receivers can be determined. The evaluation unit is thereby arranged for performing the method.
Such an apparatus can be used for providing an actual value for controlling a stabilization of a suspended load that is provided with one of both receivers and that is held by a
The precision of the method can further be improved, if linear combinations of the phase measurements are formed and if phase ambiguities thereof are searched.
For further improving the accuracy, the ratio of the wavelength to a standard deviation of the noise of the linear combination is maximized by the geometry preserving linear combination.
Additionally, it is also possible to combine the phase measurements with code measurements for increasing the wavelength of the linear combination of the phase measurements and for thereby enhancing the reliability of the estimates.
The method can be performed using an apparatus, that comprises at least two receivers, which perform code and phase measurements on navigation signals, and that is provided with an evaluation unit, by which a distance vector between the receivers can be determined. The evaluation unit is thereby arranged for performing the method.
Such an apparatus can be used for providing an actual value for controlling a stabilization of a suspended load that is provided with one of both receivers and that is held by a
7 carrier apparatus which is provided with the other one of both receivers. In particular suspended loads , that are held by a cargo helicopter or a cargo crane, can thereby be stabilized.
Further advantages and properties of the present invention are disclosed in the following description, in which embodiments of the present invention are explained in detail based on the drawing:
Figure 1 is a presentation of an apparatus for stabilizing a suspended load;
Figure 2 shows a flow diagram of a method for determining the relative position of two receivers of a satellite navigation system;
Figure 3 shows a flow diagram of a further method for determining the relative position of two receivers of a satellite navigation system;
Figure 4 is a presentation of a search tree for selecting integer phase ambiguities;
Figure 5 is a diagram, in which the probability for a wrong resolution of the phase ambiguities for various methods is depicted over several epochs;
Figure 6 is a diagram, in which the probability for a wrong resolution of the phase ambiguities is depicted over the length of the distance vector for various methods; and Figure 7 is a diagram, that shows the probability for a wrong resolution of the phase ambiguities in case of a multipath propagation.
Further advantages and properties of the present invention are disclosed in the following description, in which embodiments of the present invention are explained in detail based on the drawing:
Figure 1 is a presentation of an apparatus for stabilizing a suspended load;
Figure 2 shows a flow diagram of a method for determining the relative position of two receivers of a satellite navigation system;
Figure 3 shows a flow diagram of a further method for determining the relative position of two receivers of a satellite navigation system;
Figure 4 is a presentation of a search tree for selecting integer phase ambiguities;
Figure 5 is a diagram, in which the probability for a wrong resolution of the phase ambiguities for various methods is depicted over several epochs;
Figure 6 is a diagram, in which the probability for a wrong resolution of the phase ambiguities is depicted over the length of the distance vector for various methods; and Figure 7 is a diagram, that shows the probability for a wrong resolution of the phase ambiguities in case of a multipath propagation.
8 Figure 1 shows a satellite navigation system 1, that comprises a number of satellites 2, that are situated in an Earth's orbit. The satellites 2 emit navigation signals 3, which are modulated on carrier signals 4. The carrier signals 4 comprise in particular various wavelengths. By linear combination of the carrier signals 4, various error sources, like ionospheric errors, tropospheric errors or instrumental errors, or other error sources can be eliminated. The satellite navigation system 1 can be one of the known conventional satellite navigation systems as GPS, Galileo, Glonass, COMPASS, or any future satellite navigation system.
Figure 1 further shows a load transport system 5, that comprises an load carrier 6. The load carrier 6, for instance, can be a crane, in particular a boom of a crane, or a cargo helicopter. The load carrier 6 can be used for transporting load 7, that are attached to the load carrier 6 by means of a holding cable 8 attached to the load carrier 6. A problem associated with the transport of such suspended load 7 is that oscillations of the load 7 can be caused by side winds or by the motion of the load carrier 6. Thereby the placement of the load 7 can become difficult. The load transport system 5 is therefore provided with a load stabilization 9. The load stabilization 9 comprises a first receiver 10 attached to the load carrier 6 as well as a further receiver 11 attached to the load 7 or to the holding cable 8. The receivers 10 and 11 are respectively provided with antennas 12 for the receiving navigation signals 3. The receivers 10 and 11 can further include a signal processing unit 13.
The signal processing unit 13 processes the navigation signals 3 received by the antennas 12. This results in so called code and phase measurements, whose measured values are transferred to an evaluation unit 14 in a wire-bound or
Figure 1 further shows a load transport system 5, that comprises an load carrier 6. The load carrier 6, for instance, can be a crane, in particular a boom of a crane, or a cargo helicopter. The load carrier 6 can be used for transporting load 7, that are attached to the load carrier 6 by means of a holding cable 8 attached to the load carrier 6. A problem associated with the transport of such suspended load 7 is that oscillations of the load 7 can be caused by side winds or by the motion of the load carrier 6. Thereby the placement of the load 7 can become difficult. The load transport system 5 is therefore provided with a load stabilization 9. The load stabilization 9 comprises a first receiver 10 attached to the load carrier 6 as well as a further receiver 11 attached to the load 7 or to the holding cable 8. The receivers 10 and 11 are respectively provided with antennas 12 for the receiving navigation signals 3. The receivers 10 and 11 can further include a signal processing unit 13.
The signal processing unit 13 processes the navigation signals 3 received by the antennas 12. This results in so called code and phase measurements, whose measured values are transferred to an evaluation unit 14 in a wire-bound or
9 wireless way. Using the phase measurements or using the code and phase measurements, the evaluation unit 14 particularly calculates a distance vector 15 that represents the relative position of both receivers 10 and 11. A control unit 16, that is provided with the distance vector 15, can generate a control signal for a drive 17 based on the distance vector 15 and further actual values.
The control signal is used for moving the load carrier 6 in a manner which diminishes the oscillation of the load 7 and thereby stabilizes the load 7. In addition, it is conceivable to present the actual relative position of the load 7 and load carrier 6 on a monitor 18. The presentation on a monitor 18 can also be used to control the load carrier 6 manually instead of controlling the drive 17 by the control unit 16, or to intervene manually if there is a threatening danger.
In the embodiment shown in Figure 1, the receiver 10 is shifted with respect to receiver 11 in a lateral direction.
In practice, it will though be advantageous to attach the receiver 11 to the load carrier 6 in the region of the location, in which the holding cable 8 is attached to the load carrier 6 for facilitating the determination of the length / of the distance vector 15. If the load carrier 6 is a cargo helicopter, the length / of the distance vector 15 will be about equal to the length of the holding cable 8. In a cargo crane, the length of the holding cable 8 and therefore the length of the distance vector 15 can be determined by using a revolution counter at the drum of the holding cable 8.
For determining the distance vector 15 with sufficient accuracy, among other things the phases of the carrier signals 4 must be processed. The phases of the carrier signals 4 may, however, be affected by phase ambiguities.
In principle, it is possible to determine the phase ambiguities by a method such as the so called LAMBDA (=
Least-Squares Ambiguity Decorrelation Adjustment) method that is described in the publication by TEUNISSEN cited in the beginning. The short wavelength of the carrier signal 5 4, that is typically in the range of 19 cm, impedes a reliable determination of the integer phase ambiguities due to multipath propagation as well as unknown instrumental errors in the order of magnitude of several centimeters.
The control signal is used for moving the load carrier 6 in a manner which diminishes the oscillation of the load 7 and thereby stabilizes the load 7. In addition, it is conceivable to present the actual relative position of the load 7 and load carrier 6 on a monitor 18. The presentation on a monitor 18 can also be used to control the load carrier 6 manually instead of controlling the drive 17 by the control unit 16, or to intervene manually if there is a threatening danger.
In the embodiment shown in Figure 1, the receiver 10 is shifted with respect to receiver 11 in a lateral direction.
In practice, it will though be advantageous to attach the receiver 11 to the load carrier 6 in the region of the location, in which the holding cable 8 is attached to the load carrier 6 for facilitating the determination of the length / of the distance vector 15. If the load carrier 6 is a cargo helicopter, the length / of the distance vector 15 will be about equal to the length of the holding cable 8. In a cargo crane, the length of the holding cable 8 and therefore the length of the distance vector 15 can be determined by using a revolution counter at the drum of the holding cable 8.
For determining the distance vector 15 with sufficient accuracy, among other things the phases of the carrier signals 4 must be processed. The phases of the carrier signals 4 may, however, be affected by phase ambiguities.
In principle, it is possible to determine the phase ambiguities by a method such as the so called LAMBDA (=
Least-Squares Ambiguity Decorrelation Adjustment) method that is described in the publication by TEUNISSEN cited in the beginning. The short wavelength of the carrier signal 5 4, that is typically in the range of 19 cm, impedes a reliable determination of the integer phase ambiguities due to multipath propagation as well as unknown instrumental errors in the order of magnitude of several centimeters.
10 For improving the resolution of the phase ambiguities, previous knowledge on the length or the direction of the distance vector 15 can be used.
The vector W of the code and phase measurements can be written as:
= gd-ANd-b+ (1) With the measurements IF, the geometry matrix H, the unknown distance vector (= baseline), the wavelength matrix di, the unknown integer phase ambiguities IV, the unknown system errors b (= biases) and with the white Gaussian measurement noise n11), wherein I is the correlation matrix for the individual measurements. The problem, to find the integer phase ambiguities PJ under the constraint, that the associated distance vector comprises the known length /, can be solved by a method of least squares:
min ¨ H ¨ AN112-1 9 e R3, N E ZK with =1 (2) wherein 1: is the number of integer phase ambiguities to be determined, and thus equal to the number of carrier signals 4 multiplied by the number of satellites 2, from which the receivers 10 and 11 receive satellite signals 4.
The vector W of the code and phase measurements can be written as:
= gd-ANd-b+ (1) With the measurements IF, the geometry matrix H, the unknown distance vector (= baseline), the wavelength matrix di, the unknown integer phase ambiguities IV, the unknown system errors b (= biases) and with the white Gaussian measurement noise n11), wherein I is the correlation matrix for the individual measurements. The problem, to find the integer phase ambiguities PJ under the constraint, that the associated distance vector comprises the known length /, can be solved by a method of least squares:
min ¨ H ¨ AN112-1 9 e R3, N E ZK with =1 (2) wherein 1: is the number of integer phase ambiguities to be determined, and thus equal to the number of carrier signals 4 multiplied by the number of satellites 2, from which the receivers 10 and 11 receive satellite signals 4.
11 In the publication of GIORGI ET AL., cited in the beginning, it was proposed to linearize the distance vector. It was in particular proposed, to express the distance vector in spherical coordinates:
cos(a)cos(0) a V-y) = 1 cos(a) sin(/3) with 7 = [ 0 1 sin(a) ( 3 ) Thereby, the problem of minimizing the least squares under a constraint is transformed into a conventional problem of minimizing the least squares without constraint:
min II* ¨ HV-y)¨ AN11_1, 7 E R2, NE ZK
-y,N (4) But this problem is to a high degree non-linear.
Therefore, it was proposed to linearize the distance vector around an initial estimate (,0):
(-Y)= ("70) C(70)A7 (5) with 70 = [ao, Oor ( 6 ) and the Jacobi matrix _ ¨ sin(a0) cos(00) ¨ cos(ao) sin(00) -Ceyo) = 1 = ¨ sin(a0) sin()30) cos(a0) cos(00) cos(ao) 0 _ _ ( 7 ) The linearization results in a conventional problem of least squares:
cos(a)cos(0) a V-y) = 1 cos(a) sin(/3) with 7 = [ 0 1 sin(a) ( 3 ) Thereby, the problem of minimizing the least squares under a constraint is transformed into a conventional problem of minimizing the least squares without constraint:
min II* ¨ HV-y)¨ AN11_1, 7 E R2, NE ZK
-y,N (4) But this problem is to a high degree non-linear.
Therefore, it was proposed to linearize the distance vector around an initial estimate (,0):
(-Y)= ("70) C(70)A7 (5) with 70 = [ao, Oor ( 6 ) and the Jacobi matrix _ ¨ sin(a0) cos(00) ¨ cos(ao) sin(00) -Ceyo) = 1 = ¨ sin(a0) sin()30) cos(a0) cos(00) cos(ao) 0 _ _ ( 7 ) The linearization results in a conventional problem of least squares:
12 min ¨ HC(-yo)A-y ¨ e R2, N E ZK
^y,N
(8) with (9) Thus, the non-linearity was eliminated by a linearization with regard to a particular distance vector (/0).
The linearization of the distance vector requires a precise initial estimation of the direction of the distance vector, so that the angles of ao and ,30 can be fixed. The initial estimation is generally sufficiently precise, if the distance vector has a length greater 100 meters. In connection with the positioning of loads 7 by load carriers 6 the distances are considerably smaller, since the length of the holding cables 8 are significantly below 100 meters.
Therefore a linearization of the distance vector 15 is not possible in the present case.
In addition, the length of the distance vector 15 may vary.
The holding cable 8 may be stretched for instance, depending on the weight of the load 7.
In the following, two methods for resolving the phase ambiguities are described, which need no linearization of the distance vector 15.
A. Resolution of the Phase Ambiguities with Strong Constraints The conventional real-valued estimation N and an estimation of (-1\7) can be used for formulating the problem of minimizing the least squares as follows:
^y,N
(8) with (9) Thus, the non-linearity was eliminated by a linearization with regard to a particular distance vector (/0).
The linearization of the distance vector requires a precise initial estimation of the direction of the distance vector, so that the angles of ao and ,30 can be fixed. The initial estimation is generally sufficiently precise, if the distance vector has a length greater 100 meters. In connection with the positioning of loads 7 by load carriers 6 the distances are considerably smaller, since the length of the holding cables 8 are significantly below 100 meters.
Therefore a linearization of the distance vector 15 is not possible in the present case.
In addition, the length of the distance vector 15 may vary.
The holding cable 8 may be stretched for instance, depending on the weight of the load 7.
In the following, two methods for resolving the phase ambiguities are described, which need no linearization of the distance vector 15.
A. Resolution of the Phase Ambiguities with Strong Constraints The conventional real-valued estimation N and an estimation of (-1\7) can be used for formulating the problem of minimizing the least squares as follows:
13 min N - N + min (N)-(N) (10) (N) ) Thus, the one integer phase ambiguities Nand the one distance vector &Iõ are searched, which minimizes the evaluation function, which is given in brackets in formula (10).
The real-valued ambiguities are obtained by an orthogonal projection of the measurements T on the space of II:
H
PIT=PIAN+P B+P/7 with 11HI H .1_ 7"7" =ki (11) A
An estimation according to the least square method then results in a real-valued estimation of the phase ambiguities:
1sT = (ATE-1A)-1ATE-141 (12) Similarly the distance vector 7-) is obtained by an estimation according to the least square method:
4-(N) = (HT E-1 H)_i HT E-1H(41 - AN) (13) The second term of the evaluation function which has to be minimized according to expression (10) can also be presented as Lagrange function with the Lagrange parameter A:
30f(2) = (N)- +2( (N) _12)
The real-valued ambiguities are obtained by an orthogonal projection of the measurements T on the space of II:
H
PIT=PIAN+P B+P/7 with 11HI H .1_ 7"7" =ki (11) A
An estimation according to the least square method then results in a real-valued estimation of the phase ambiguities:
1sT = (ATE-1A)-1ATE-141 (12) Similarly the distance vector 7-) is obtained by an estimation according to the least square method:
4-(N) = (HT E-1 H)_i HT E-1H(41 - AN) (13) The second term of the evaluation function which has to be minimized according to expression (10) can also be presented as Lagrange function with the Lagrange parameter A:
30f(2) = (N)- +2( (N) _12)
(14) EE-0v) The minimization of PA) with respect to 42(V) results in an estimation for the distance vector 15 as a function of the Lagrange parameter A and the integer phase ambiguities N:
2(N) = (N) (N)E71 ¨ 21) E-' (N)
2(N) = (N) (N)E71 ¨ 21) E-' (N)
(15) ( This estimation of the distance vector of length / can be introduced in the constraint for the distance:
, (N) E7.1 _21)-1 ETI (N) ( 2 N) _12 = 0
, (N) E7.1 _21)-1 ETI (N) ( 2 N) _12 = 0
(16) ¶.5 The unknown in equation (16) are the Lagrange parameter A
and the integer phase ambiguities N. This equation must be resolved for the Lagrange parameter, which cannot be done by an analytic expression. However, by using a multidimensional Gauss-Newton-method, equation (16), can efficiently be solved.
For each candidate of the integer vector of the phase ambiguities N the Lagrange parameter A is determined iteratively:
An+i = + J-1(2,75 N) = f (2õ)
and the integer phase ambiguities N. This equation must be resolved for the Lagrange parameter, which cannot be done by an analytic expression. However, by using a multidimensional Gauss-Newton-method, equation (16), can efficiently be solved.
For each candidate of the integer vector of the phase ambiguities N the Lagrange parameter A is determined iteratively:
An+i = + J-1(2,75 N) = f (2õ)
(17) with the Jacobi matrix:
N)=af(2,N)
N)=af(2,N)
(18) aA, and the Lagrange function:
fRoNM= (N) (N)E71 _21) E71 (N) -12 ( n
fRoNM= (N) (N)E71 _21) E71 (N) -12 ( n
(19) The initial value for / is chosen to be 0. The iterative calculation of 2 is performed as long as the value of 2 has converged.
5 Figure 2 illustrates the flow of the method: In a first step, the phase measurements 19 are performed on the carrier signals 2. In parallel, the code measurements 20 can also be performed. By a linear combination 21, combination signals can be formed from the phase measured 10 values and code measured values obtained from the phase measurements 19 and code measurements 20. The combination signals can also be formed by the phase measurements only.
For this combination signals an estimation 22 according to equation (12) is performed for the phase ambiguities Ar, 15 which results in real-valued phase ambiguities N. As a next step, a limitation 23 of the considered integer phase ambiguities is made depending on an integrity factor K, and a number of candidates /V is determined for the integer phase ambiguities Pion the basis of the real-valued phase ambiguities. A method that can be used for the limitation 23 is described in more details in the following. Among all considered integer phase ambiguities a particular integer phase ambiguity is selected by a selection 24. By a further estimation 25 according to equation (13), the distance vector (N) is estimated for the selected integer phase ambiguities. Afterwards an iterative calculation 26 of A according to formulas (17) to (19) is performed. After 2 has been converged, a calculation 27 of 42(V) according to formula (15) is carried out. An evaluation 28 of the evaluation function according to formula (10) will follow. If the evaluation function has not yet been evaluated for all candidates /V, a candidate for the integer phase ambiguities is again selected by the selection 25, and the subsequent method steps are repeated until the calculation 28 of the evaluation function. When the calculation 28 has been performed for all candidates Ar for the integer phase ambiguities, the one vector AT of the integer phase ambiguities, for which the evaluation function according to formula (10) yields the smallest value, can be selected by _ a final selection 29. The associated distance vector 2(N) can then be provided to the control unit 15 as an actual value.
B. Resolving the Phase Ambiguities with Weak Constraints In this method, the distance vector 15 need not necessarily have the length L This is achieved in the evaluation function by coupling the corresponding constraint only weakly with the coupling parameter P.
In the present case, the evaluation function can be written as:
J ( , N) = tli - IR ANIL, + ,u = ( _l)2 (20) In this evaluation function, the first term represents the weighted sum of squared errors (SSE) and the second term represents the difference between the length of the distance vector and a previously known length 1 of the distance vector 15. The parameter A depends on the reliability of the previous knowledge. A big value for A
indicates that the length value / is considered as reliable, whereas a low value for A indicates, that the length value 1 is considered as erroneous. There is an essential difference between the coupling parameter A and the Lagrange parameter A of the preceding method. In the preceding method the Lagrange parameter A was calculated iteratively for minimizing the evaluation function. The coupling parameter A, however, is a priori set to express the strength of the constraint. In the preceding method, the found distance vector comprises exactly the length 1, wherein in the present method the found distance vector may have a length, which deviates from the length value 1.
The derivation of the evaluation function with respect to the distance vector results in:
aJ
¨ = -2HT E-1(T - AN)+2HT 1-11R + ,u-1101 2( 4:1 - (21) a4=
This equation cannot be solved for the distance vector in closed form. The solution, however, can be found iteratively using the Newton method of the steepest gradient:
j2 OJ
n-F1 = C't a^2 (22) with the second derivation of the evaluation function:
\\
a j2 gT
=2HTE-1H+2p= (23) 4a2, As an initial value the estimation for according to equation (13) can be used.
The flow of the method is illustrated in Figure 3: The method starts with the same method steps as the method illustrated in Figure 2. After the selection 25 an iterative solution 30 of equation (21) is performed, wherein the distance vector i(A) estimated in the estimation 23 is used as initial value. The remaining method steps correspond again to the method steps of the method illustrated in Figure 2 with the exception that a depending on the coupling parameter p is calculated instead of a A depending on A.
C. Limitation of the Integer Phase Ambiguities In the methods illustrated in Figures 2 and 3, before the selection 24 of the integer phase ambiguities, the limitation 23 is performed, which is described in detail in the following.
The search for candidates for the integer phase ambiguities aims at finding an integer candidate vector /V which fulfils the condition:
= iS7112 .67 (24) In practice the search space is limited by a given value:
11N¨ Ar112-1 < X2 E ¨
icr (25) wherein X2 can be chosen to be equal to the error norm of the sequential solution without constraints. Thus, it is possible, to limit the search for candidates for the integer phase ambiguities.
As described in the publications of TEUNISSEN mentioned in the beginning, the error norm can also be expressed by estimations for the integer errors as:
(Ni 11N - S r 11 - 1 = 2 N11 i-i (28) with k-1 = -1\4 Ea ¨-2 (g=11 = 1 ¨ [1Cr=l1 = 1]) j=1 (29) and k-1 cr,c,0_2 _ \--"` c-k-1 iST'k NN1 j=1 (30) wherein H in equation (29) refers to a truncation to the nearest integer value.
The squared sum of the conditioned errors is now transformed, wherein one term is kept on the left side and the remaining terms are transferred to the right side.
This results in:
i-1 (N/ / 1)2 (Ni ¨ / 1)2 ¨
< X2 E __________________________________________________ 2 ____________________________________ 2 2 1=1 /=1+1 i-1 ( 2 N1 Ars /11,.. / 1 < X2 ¨
1=, (31) what enables an sequential search for the integer phase ambiguities. The set of candidates for each integer phase ambiguity Ni is therefore given by:
i-1 (N/ / 1 2 Cig x2 1=1 (32) what provides a lower and an upper limit for Thus, a sequential search tree for all components of the vector of integer phase ambiguities having a error norm smaller than 2 can be constructed. The efficiency of the search can further be improved by checking the length of the distance vector in each branch of the search tree. A branch of the 5 search tree is only prosecuted, if the difference between the estimated length of the distance vector 15 and its previously known value is smaller than the standard deviation of the length estimation of the distance vector multiplied by an integrity factor K:
HIE(Nmi)11-11 5-- = C111411 (33) with the estimation for the length of the distance vector without constraints:
(Nmi) = arg min II* ¨ kg¨ ANm,111-1 (34) and the partially fixed vector of the phase ambiguities Nm-, = [ATI ... gm, I gm, 1 NK1 (35) in which the first Afi phase ambiguities have already been set to integer values and the remaining components are still unknown. It is obvious, that the increasing number of fixed phase ambiguities results in a smaller standard deviation of the distance estimation. In consequence, the distance limitation is becoming stronger with increasing what results in an essential improvement of the search efficiency.
The standard deviation of the length estimation in formula (33) can further be calculated by:
a = Vcr? ______________ cr?
(36) Figure 3 shows an example of a search tree 31. The search tree comprises an initial node 32, to which is associated with a possibility for selecting the first component of the vector /IV of the candidates for the integer phase ambiguities. The initial node 32 is followed in downward direction by various nodal planes 33, from where nodes 34 can be selected for further components of the vectors of phase ambiguities. Final nodes 35 are disposed on the lowest nodal plane 33, in which the last component of the vector ST of the phase ambiguities is fixed. The search tree 31 therefore comprises various search tree branches 36, which extend respectively from the initial nodes 32 to the final nodes 35, and which correspond to the various possibilities to set the vector AT' of the phase ambiguities.
The search tree is now processed as follows:
In a first step, the lower and the upper limit and are determined, which can generally expressed as:
/-= v . ,i-1 %
= 0 -2 (37) N
= N r N X2 -(38) 1=1 Is 1 m, The first component /V1 is determined within the limits and < N < u - r (39) In the example shown in Figure 4, there are three possibilities for allocating ST,, since three search tree branches 36 originate from initial node 32. Subsequently, the three search tree branches 36 are checked for the condition given in formula (33). In the present case, it is particularly checked whether (NM) - Ka(N) (40) is valid. The search tree branch 36 is only prosecuted, if the condition (40) is met. In the present case only the outer two search tree branches 36 are prosecuted.
In the next nodal plane 36, the integer values for the component ST2 are selected, where for Sr2 shall apply:
N < N2 <u (39) ^õ,r fir211 The selected four values of ST2 are then checked again, whether the condition (33) is fulfilled.
(NM)-1 - ( 41) 2 A,f2 In the example illustrated in Figure 4, only the two inner search tree branches 36 are then maintained.
In the following nodal plane 33, the values for the component Spi3 are selected, and subsequently the selected values are again checked with respect to the condition (33).
In the nodal plane 33 above the final nodes 35, the values for ./Vic are finally selected, wherein:
< NI( <UAr ............................... K¨I
(42) = ,K¨I Ku The selected values 81K are checked again thereupon, whether the selected values i\--Tic are fulfilling the condition (33):
(Nivi) ¨/
(43) j(Armic) In the embodiment shown in Figure 4, only three search tree branches 36 arrive at the final nodes 35. Insofar, only three candidates for the integer phase ambiguities must be evaluated.
D. Further Improvements with Multifrequency Combinations of Code and Phase Measurements The reliability of the resolving the phase ambiguities can be further improved by multifrequency linear combinations having long wavelength. The combinations can include code ,k measurements l'u,rn -t and phase measurements Am*k :
EarnArnouk,rn + ornpuk,.
m=i (44) ,k with the combined wavelength A, the phase coefficients 'I/071 Rk and the code coefficients -'14,rn. The coefficients are selected such that the phase ambiguities can maximally be distinguished:
A
max D= max am,i3m am,o,,, 2a (45) wherein further conditions must be fulfilled. Firstly, the linear combination must preserve geometry:
E +Om = 1 m=1 (46) and secondly, the linear combination of the phase ambiguities of different wavelengths must be an integer multiple of the combined wavelength:
EaAN = AN
m=1 (47) which is equivalent to:
N=Ea-AmNn, A
Jm (48) Since Arm is unknown, but an integer number, jm must be an integer number for obtaining an integer N. Transforming this equation results in the phase coefficients:
am = in-tA
Am (49) which depends on the integer coefficients jm and the combined wavelength M.
wo A = _________________ ER
m=i (50) wherein CO0 is the combined phase coefficient:
Wçb =E am.
m=1 (51) The multipath propagation and the instrumental errors can often be recognized by deviations of the distance vector 15 5 and the weighted sum of the squared errors. Some instrumental errors, however, can not be recognized by means of distance deviations. If the integer phase ambiguities are estimated by minimizing least squares without constraints the following types of instrumental 10 errors cannot be detected:
bl = -11,6, and b2 = AAN (52) wherein the first vector of the instrumental errors 15 corresponds to a positional deviation and the second vector of the instrumental errors corresponds to a deviation of the integer phase ambiguities. Both cases are relatively unlikely. The resolution of the phase ambiguities and the constraints reduces the set of instrumental errors that
5 Figure 2 illustrates the flow of the method: In a first step, the phase measurements 19 are performed on the carrier signals 2. In parallel, the code measurements 20 can also be performed. By a linear combination 21, combination signals can be formed from the phase measured 10 values and code measured values obtained from the phase measurements 19 and code measurements 20. The combination signals can also be formed by the phase measurements only.
For this combination signals an estimation 22 according to equation (12) is performed for the phase ambiguities Ar, 15 which results in real-valued phase ambiguities N. As a next step, a limitation 23 of the considered integer phase ambiguities is made depending on an integrity factor K, and a number of candidates /V is determined for the integer phase ambiguities Pion the basis of the real-valued phase ambiguities. A method that can be used for the limitation 23 is described in more details in the following. Among all considered integer phase ambiguities a particular integer phase ambiguity is selected by a selection 24. By a further estimation 25 according to equation (13), the distance vector (N) is estimated for the selected integer phase ambiguities. Afterwards an iterative calculation 26 of A according to formulas (17) to (19) is performed. After 2 has been converged, a calculation 27 of 42(V) according to formula (15) is carried out. An evaluation 28 of the evaluation function according to formula (10) will follow. If the evaluation function has not yet been evaluated for all candidates /V, a candidate for the integer phase ambiguities is again selected by the selection 25, and the subsequent method steps are repeated until the calculation 28 of the evaluation function. When the calculation 28 has been performed for all candidates Ar for the integer phase ambiguities, the one vector AT of the integer phase ambiguities, for which the evaluation function according to formula (10) yields the smallest value, can be selected by _ a final selection 29. The associated distance vector 2(N) can then be provided to the control unit 15 as an actual value.
B. Resolving the Phase Ambiguities with Weak Constraints In this method, the distance vector 15 need not necessarily have the length L This is achieved in the evaluation function by coupling the corresponding constraint only weakly with the coupling parameter P.
In the present case, the evaluation function can be written as:
J ( , N) = tli - IR ANIL, + ,u = ( _l)2 (20) In this evaluation function, the first term represents the weighted sum of squared errors (SSE) and the second term represents the difference between the length of the distance vector and a previously known length 1 of the distance vector 15. The parameter A depends on the reliability of the previous knowledge. A big value for A
indicates that the length value / is considered as reliable, whereas a low value for A indicates, that the length value 1 is considered as erroneous. There is an essential difference between the coupling parameter A and the Lagrange parameter A of the preceding method. In the preceding method the Lagrange parameter A was calculated iteratively for minimizing the evaluation function. The coupling parameter A, however, is a priori set to express the strength of the constraint. In the preceding method, the found distance vector comprises exactly the length 1, wherein in the present method the found distance vector may have a length, which deviates from the length value 1.
The derivation of the evaluation function with respect to the distance vector results in:
aJ
¨ = -2HT E-1(T - AN)+2HT 1-11R + ,u-1101 2( 4:1 - (21) a4=
This equation cannot be solved for the distance vector in closed form. The solution, however, can be found iteratively using the Newton method of the steepest gradient:
j2 OJ
n-F1 = C't a^2 (22) with the second derivation of the evaluation function:
\\
a j2 gT
=2HTE-1H+2p= (23) 4a2, As an initial value the estimation for according to equation (13) can be used.
The flow of the method is illustrated in Figure 3: The method starts with the same method steps as the method illustrated in Figure 2. After the selection 25 an iterative solution 30 of equation (21) is performed, wherein the distance vector i(A) estimated in the estimation 23 is used as initial value. The remaining method steps correspond again to the method steps of the method illustrated in Figure 2 with the exception that a depending on the coupling parameter p is calculated instead of a A depending on A.
C. Limitation of the Integer Phase Ambiguities In the methods illustrated in Figures 2 and 3, before the selection 24 of the integer phase ambiguities, the limitation 23 is performed, which is described in detail in the following.
The search for candidates for the integer phase ambiguities aims at finding an integer candidate vector /V which fulfils the condition:
= iS7112 .67 (24) In practice the search space is limited by a given value:
11N¨ Ar112-1 < X2 E ¨
icr (25) wherein X2 can be chosen to be equal to the error norm of the sequential solution without constraints. Thus, it is possible, to limit the search for candidates for the integer phase ambiguities.
As described in the publications of TEUNISSEN mentioned in the beginning, the error norm can also be expressed by estimations for the integer errors as:
(Ni 11N - S r 11 - 1 = 2 N11 i-i (28) with k-1 = -1\4 Ea ¨-2 (g=11 = 1 ¨ [1Cr=l1 = 1]) j=1 (29) and k-1 cr,c,0_2 _ \--"` c-k-1 iST'k NN1 j=1 (30) wherein H in equation (29) refers to a truncation to the nearest integer value.
The squared sum of the conditioned errors is now transformed, wherein one term is kept on the left side and the remaining terms are transferred to the right side.
This results in:
i-1 (N/ / 1)2 (Ni ¨ / 1)2 ¨
< X2 E __________________________________________________ 2 ____________________________________ 2 2 1=1 /=1+1 i-1 ( 2 N1 Ars /11,.. / 1 < X2 ¨
1=, (31) what enables an sequential search for the integer phase ambiguities. The set of candidates for each integer phase ambiguity Ni is therefore given by:
i-1 (N/ / 1 2 Cig x2 1=1 (32) what provides a lower and an upper limit for Thus, a sequential search tree for all components of the vector of integer phase ambiguities having a error norm smaller than 2 can be constructed. The efficiency of the search can further be improved by checking the length of the distance vector in each branch of the search tree. A branch of the 5 search tree is only prosecuted, if the difference between the estimated length of the distance vector 15 and its previously known value is smaller than the standard deviation of the length estimation of the distance vector multiplied by an integrity factor K:
HIE(Nmi)11-11 5-- = C111411 (33) with the estimation for the length of the distance vector without constraints:
(Nmi) = arg min II* ¨ kg¨ ANm,111-1 (34) and the partially fixed vector of the phase ambiguities Nm-, = [ATI ... gm, I gm, 1 NK1 (35) in which the first Afi phase ambiguities have already been set to integer values and the remaining components are still unknown. It is obvious, that the increasing number of fixed phase ambiguities results in a smaller standard deviation of the distance estimation. In consequence, the distance limitation is becoming stronger with increasing what results in an essential improvement of the search efficiency.
The standard deviation of the length estimation in formula (33) can further be calculated by:
a = Vcr? ______________ cr?
(36) Figure 3 shows an example of a search tree 31. The search tree comprises an initial node 32, to which is associated with a possibility for selecting the first component of the vector /IV of the candidates for the integer phase ambiguities. The initial node 32 is followed in downward direction by various nodal planes 33, from where nodes 34 can be selected for further components of the vectors of phase ambiguities. Final nodes 35 are disposed on the lowest nodal plane 33, in which the last component of the vector ST of the phase ambiguities is fixed. The search tree 31 therefore comprises various search tree branches 36, which extend respectively from the initial nodes 32 to the final nodes 35, and which correspond to the various possibilities to set the vector AT' of the phase ambiguities.
The search tree is now processed as follows:
In a first step, the lower and the upper limit and are determined, which can generally expressed as:
/-= v . ,i-1 %
= 0 -2 (37) N
= N r N X2 -(38) 1=1 Is 1 m, The first component /V1 is determined within the limits and < N < u - r (39) In the example shown in Figure 4, there are three possibilities for allocating ST,, since three search tree branches 36 originate from initial node 32. Subsequently, the three search tree branches 36 are checked for the condition given in formula (33). In the present case, it is particularly checked whether (NM) - Ka(N) (40) is valid. The search tree branch 36 is only prosecuted, if the condition (40) is met. In the present case only the outer two search tree branches 36 are prosecuted.
In the next nodal plane 36, the integer values for the component ST2 are selected, where for Sr2 shall apply:
N < N2 <u (39) ^õ,r fir211 The selected four values of ST2 are then checked again, whether the condition (33) is fulfilled.
(NM)-1 - ( 41) 2 A,f2 In the example illustrated in Figure 4, only the two inner search tree branches 36 are then maintained.
In the following nodal plane 33, the values for the component Spi3 are selected, and subsequently the selected values are again checked with respect to the condition (33).
In the nodal plane 33 above the final nodes 35, the values for ./Vic are finally selected, wherein:
< NI( <UAr ............................... K¨I
(42) = ,K¨I Ku The selected values 81K are checked again thereupon, whether the selected values i\--Tic are fulfilling the condition (33):
(Nivi) ¨/
(43) j(Armic) In the embodiment shown in Figure 4, only three search tree branches 36 arrive at the final nodes 35. Insofar, only three candidates for the integer phase ambiguities must be evaluated.
D. Further Improvements with Multifrequency Combinations of Code and Phase Measurements The reliability of the resolving the phase ambiguities can be further improved by multifrequency linear combinations having long wavelength. The combinations can include code ,k measurements l'u,rn -t and phase measurements Am*k :
EarnArnouk,rn + ornpuk,.
m=i (44) ,k with the combined wavelength A, the phase coefficients 'I/071 Rk and the code coefficients -'14,rn. The coefficients are selected such that the phase ambiguities can maximally be distinguished:
A
max D= max am,i3m am,o,,, 2a (45) wherein further conditions must be fulfilled. Firstly, the linear combination must preserve geometry:
E +Om = 1 m=1 (46) and secondly, the linear combination of the phase ambiguities of different wavelengths must be an integer multiple of the combined wavelength:
EaAN = AN
m=1 (47) which is equivalent to:
N=Ea-AmNn, A
Jm (48) Since Arm is unknown, but an integer number, jm must be an integer number for obtaining an integer N. Transforming this equation results in the phase coefficients:
am = in-tA
Am (49) which depends on the integer coefficients jm and the combined wavelength M.
wo A = _________________ ER
m=i (50) wherein CO0 is the combined phase coefficient:
Wçb =E am.
m=1 (51) The multipath propagation and the instrumental errors can often be recognized by deviations of the distance vector 15 5 and the weighted sum of the squared errors. Some instrumental errors, however, can not be recognized by means of distance deviations. If the integer phase ambiguities are estimated by minimizing least squares without constraints the following types of instrumental 10 errors cannot be detected:
bl = -11,6, and b2 = AAN (52) wherein the first vector of the instrumental errors 15 corresponds to a positional deviation and the second vector of the instrumental errors corresponds to a deviation of the integer phase ambiguities. Both cases are relatively unlikely. The resolution of the phase ambiguities and the constraints reduces the set of instrumental errors that
20 cannot be recognized because:
(53) 25 E. Advantages The resolution of the phase ambiguities and the constraints has two essential advantages in comparison with a phase resolution without constraints. On the one hand, the resolution of the phase ambiguities becomes more reliable, and, on the other hand, the estimation of the distance vector becomes more accurate.
Figure 5 illustrates the advantage of resolving the phase ambiguities under constraints using phase ambiguities of an E1-E5 linear combination of double difference measurements as an example, wherein only phase measurements with a combined wavelength of 2\ = 78 cm have been processed. The exact knowledge on the length of the distance vector reduces the error probability for a wrong determination of the phase ambiguities by several orders of magnitudes.
The error probability Pwf shown in Figure 5 is defined by:
K {1, if Nk #N
P = P(N # N) = liml f (N K) with f (N k) =
wf k-300 k=1 0, otherwise (54) and is determined by extensive Monte-Carlo simulations based on the statistics of the estimation for the real-valued phase ambiguities:
N =N (N , EA,. ) (55) In the Monte-Carlo simulations, the vector of the true ambiguities AT is initially set to an arbitrary value.
Afterwards, various real-valued phase ambiguities are generated according to their probability distribution and the method depicted in Figure 2 and 3 is applied. After each passage through the method, it is determined, whether the phase ambiguities have been resolved correctly. By averaging over the results of the Monte-Carlo simulation the error probability is then calculated.
It can be recognized from Figure 6 that resolving the phase ambiguities with weak constraints is considerably better than the other methods, while the length of the distance vector 15 is not known from the outset. By resolving the phase ambiguities with weak constraints, the search space is limited, which results in a lower error rate in comparison to resolving without constraints. Resolving with weak constraint is also better than resolving with strong constraint, since resolving with strong constraint restricts the search space too severely. The integer candidates for both resolutions with constraints where selected among the ten best candidates, that had been determined using the LAMBDA method. It can be recognized from Figure 6 that the coupling factor A should be selected according to the expected precision of the predetermined value.
Figure 7 demonstrates that the resolution with weak constraints is also advantageous in case of a stationary multipath propagation. The diagram of Figure 7 was generated by simulating the resolution of the phase ambiguities in the presence of an instrumental error, which was equivalent to multipath propagation and which had the indicated standard deviation of a mean value free Gaussian distribution.
The method described herein offers the possibility of stabilizing the load 7 by determining the distance vector 15 between both receivers 10 and 11. In contrast to conventional methods, this is also possible at cable lengths below 100 meters.
Besides pure phase combinations, also code and phase combinations can be used, for instance, an optimized E1-E5 linear combination with a combined wavelength of 3.285 m, a noise level in the range of few centimeters and a suppression of multipath errors of 12.6 dB.
It should further be noted that phase measurements can also be difference or double difference measurements.
The method described herein can be used for determining one single relative position. In a modified embodiment, several relative distances are determined, between several receivers, either stationary or attached to the load carrier, and a further receiver in the vicinity of the suspended load. Based on the determined relative positions and the known relative position of the receivers, that are stationary or attached to the load carrier, the position of the receivers can be checked for consistency by triangulation or the errors can be eliminated by averaging.
It should finally be pointed out that features and characteristics described in conjunction with a particular aspect, embodiment or example of the invention are to be understood to be applicable to any other aspect, embodiment or example described herein unless incompatible therewith.
It should further be pointed out in the end that throughout the description and claims of this specification, the singular encompasses the plural unless the context otherwise requires. In particular, where the indefinite article is used, the specification is to be understood as contemplating plurality as well as singularity, unless the context requires otherwise.
(53) 25 E. Advantages The resolution of the phase ambiguities and the constraints has two essential advantages in comparison with a phase resolution without constraints. On the one hand, the resolution of the phase ambiguities becomes more reliable, and, on the other hand, the estimation of the distance vector becomes more accurate.
Figure 5 illustrates the advantage of resolving the phase ambiguities under constraints using phase ambiguities of an E1-E5 linear combination of double difference measurements as an example, wherein only phase measurements with a combined wavelength of 2\ = 78 cm have been processed. The exact knowledge on the length of the distance vector reduces the error probability for a wrong determination of the phase ambiguities by several orders of magnitudes.
The error probability Pwf shown in Figure 5 is defined by:
K {1, if Nk #N
P = P(N # N) = liml f (N K) with f (N k) =
wf k-300 k=1 0, otherwise (54) and is determined by extensive Monte-Carlo simulations based on the statistics of the estimation for the real-valued phase ambiguities:
N =N (N , EA,. ) (55) In the Monte-Carlo simulations, the vector of the true ambiguities AT is initially set to an arbitrary value.
Afterwards, various real-valued phase ambiguities are generated according to their probability distribution and the method depicted in Figure 2 and 3 is applied. After each passage through the method, it is determined, whether the phase ambiguities have been resolved correctly. By averaging over the results of the Monte-Carlo simulation the error probability is then calculated.
It can be recognized from Figure 6 that resolving the phase ambiguities with weak constraints is considerably better than the other methods, while the length of the distance vector 15 is not known from the outset. By resolving the phase ambiguities with weak constraints, the search space is limited, which results in a lower error rate in comparison to resolving without constraints. Resolving with weak constraint is also better than resolving with strong constraint, since resolving with strong constraint restricts the search space too severely. The integer candidates for both resolutions with constraints where selected among the ten best candidates, that had been determined using the LAMBDA method. It can be recognized from Figure 6 that the coupling factor A should be selected according to the expected precision of the predetermined value.
Figure 7 demonstrates that the resolution with weak constraints is also advantageous in case of a stationary multipath propagation. The diagram of Figure 7 was generated by simulating the resolution of the phase ambiguities in the presence of an instrumental error, which was equivalent to multipath propagation and which had the indicated standard deviation of a mean value free Gaussian distribution.
The method described herein offers the possibility of stabilizing the load 7 by determining the distance vector 15 between both receivers 10 and 11. In contrast to conventional methods, this is also possible at cable lengths below 100 meters.
Besides pure phase combinations, also code and phase combinations can be used, for instance, an optimized E1-E5 linear combination with a combined wavelength of 3.285 m, a noise level in the range of few centimeters and a suppression of multipath errors of 12.6 dB.
It should further be noted that phase measurements can also be difference or double difference measurements.
The method described herein can be used for determining one single relative position. In a modified embodiment, several relative distances are determined, between several receivers, either stationary or attached to the load carrier, and a further receiver in the vicinity of the suspended load. Based on the determined relative positions and the known relative position of the receivers, that are stationary or attached to the load carrier, the position of the receivers can be checked for consistency by triangulation or the errors can be eliminated by averaging.
It should finally be pointed out that features and characteristics described in conjunction with a particular aspect, embodiment or example of the invention are to be understood to be applicable to any other aspect, embodiment or example described herein unless incompatible therewith.
It should further be pointed out in the end that throughout the description and claims of this specification, the singular encompasses the plural unless the context otherwise requires. In particular, where the indefinite article is used, the specification is to be understood as contemplating plurality as well as singularity, unless the context requires otherwise.
Claims (18)
1. Method for determining the relative position between two receivers (10, 11) for satellite navigation, in which phase measurements on carrier signals (4) of a satellite navigation system (1) are performed by the receivers (10, 11), and in which the relative position of the receivers (10, 11) is determined by an evaluation unit (14) on the basis of the phase measurements, characterized in that - integer phase ambiguities and a distance vector (15) describing the relative position are determined by the evaluation unit (14), by - minimizing an evaluation function, which comprises besides a first measure for evaluating the consistency of the integer phase ambiguities with the phase measurements, a second measure, that determines the deviation of the distance vector (15), which describes the relative position and is associated with the integer phase ambiguity, from a distance vector of predetermined length.
2. Method according to claim 1, - in which the real-valued phase ambiguities are initially determined and then the integer phase ambiguities and the first distance vector (15) describing the relative position is determined by optimizing the evaluation function, - in which the first measure for evaluating the integer phase ambiguities comprises a norm, that determines the deviation of the real-valued phase ambiguities from the integer phase ambiguities, and - in which the second measure comprises a norm, which determines the deviation of the distance vector describing the relative position and associated with the integer phase ambiguities from the distance vector of predetermined length.
3. Method according to claim 2, - in which the distance vector of predetermined length is determined by searching the one distance vector, which has a predetermined length and the smallest distance to the distance vector describing the relative position and associated with the integer phase ambiguities, by means of a search evaluation function.
4. Method according to claim 3, in which the search evaluation function is a Lagrange function.
5. Method according to claim 4, in which the evaluation function is given by:
and in which the search evaluation function is given by:
with:
N: the vector of the integer phase ambiguities, for which the evaluation function is optimized, .xi..lambda.(N): the searched distance vector, N: the vector of the estimated real-valued phase ambiguities, .xi.(N): the distance vector estimated on the basis of N
i: the previously known length of the searched distance vector (15), .lambda. : the Lagrange factor and .SIGMA.: the respective covariance matrixes.
and in which the search evaluation function is given by:
with:
N: the vector of the integer phase ambiguities, for which the evaluation function is optimized, .xi..lambda.(N): the searched distance vector, N: the vector of the estimated real-valued phase ambiguities, .xi.(N): the distance vector estimated on the basis of N
i: the previously known length of the searched distance vector (15), .lambda. : the Lagrange factor and .SIGMA.: the respective covariance matrixes.
6. Method according to claim 1, - in which the first measure for evaluating the integer phase ambiguities comprises a norm that determines the deviation of calculated phase values, which have been calculated based on integer phase ambiguities and the distance vector, from measured phase values, which have been detected by code and phase measurements, and - in which the second measure determines the deviation of a norm of the distance vector from a predetermined length.
7. Method according to claim 6, in which the first measure is connected to the second measure by a coupling parameter that adjusts the influence of the second measure on the determination of the integer phase ambiguities and of the distance vector.
8. Method according to claim 7, in which the evaluation function is given by:
with:
N: the vector of the integer phase ambiguities, for which the evaluation function is optimized, .xi.: the searched distance vector, .PSI.: the vector of the measured values H: the geometry matrix A: the wavelength matrix, µ : the coupling factor l: the predetermined length of the searched distance vector (15) and .SIGMA.:the respective covariance matrix.
with:
N: the vector of the integer phase ambiguities, for which the evaluation function is optimized, .xi.: the searched distance vector, .PSI.: the vector of the measured values H: the geometry matrix A: the wavelength matrix, µ : the coupling factor l: the predetermined length of the searched distance vector (15) and .SIGMA.:the respective covariance matrix.
9. Method according to any one of claims 1 to 8, in which the evaluation function is repeatedly evaluated for various integer phase ambiguities.
10. Method according to claim 9, in which the phase ambiguities to be evaluated are searched by means of a search tree, whose branches correspond to series of phase ambiguities and in which search tree branches with a phase ambiguity or a length of the distance vector beyond probability limits are excluded.
11. Method according to claim 10, in which the probability limits are determined by the error norm of the estimation of the real-valued phase ambiguities, and in which a branch of the search tree is only prosecuted, if the difference between the estimated length of the distance vector (15) and its previously known value is smaller than the standard deviation of the length estimation of the distance vector (15) multiplied by a factor.
12. Method according to any one of claims 1 to 11, in which at least one linear combination of phase measurements is used, and in which the phase ambiguities of the linear combination and the distance vector are searched by evaluating the evaluation function.
13. Method according to claim 12, in which the wavelength of the phase measurements is increased by at least one geometry preserving linear combination.
14. Method according to claim 13, in which the ratio of the wavelength to the standard deviation of the noise of the linear combination is maximized by the at least one geometry preserving linear combination.
15. Method according to any one of claims 12 to 14, in which the linear combination includes code measurements.
16. Apparatus for determining a relative position between at least two user devices comprising:
- at least two receivers (10, 11) which are disposed at a distance and which perform code and phase measurements on navigation signals, which the receivers (10, 11) receive from satellites (2) of a satellite navigation system (1), and - an evaluation unit (14), which can determine a distance vector (15) between the receivers (10, 11) on the basis of the code and phase measurements, characterized in that the receivers (10, 11) and the evaluation unit (14) are arranged for performing the method according to any one of claims 1 to 15.
- at least two receivers (10, 11) which are disposed at a distance and which perform code and phase measurements on navigation signals, which the receivers (10, 11) receive from satellites (2) of a satellite navigation system (1), and - an evaluation unit (14), which can determine a distance vector (15) between the receivers (10, 11) on the basis of the code and phase measurements, characterized in that the receivers (10, 11) and the evaluation unit (14) are arranged for performing the method according to any one of claims 1 to 15.
17. Use of the apparatus according to claim 16, characterized in that the distance vector determined by the apparatus is used as an actual value for a control unit (15) for stabilizing a suspended load (7), which is provided with one of both receivers (10) and which is held by a carrier apparatus (6), which is provided with the other one of both receivers (11).
18. Use according to claim 17, in which the carrier apparatus (6) is a cargo helicopter or a cargo crane.
Applications Claiming Priority (3)
| Application Number | Priority Date | Filing Date | Title |
|---|---|---|---|
| DE102010038257 | 2010-10-18 | ||
| DE102010038257.4 | 2010-10-18 | ||
| PCT/EP2011/067554 WO2012052307A1 (en) | 2010-10-18 | 2011-10-07 | Method and apparatus for determining the relative position between two receivers and use of the apparatus for stabilizing suspended loads |
Publications (2)
| Publication Number | Publication Date |
|---|---|
| CA2814882A1 CA2814882A1 (en) | 2012-04-26 |
| CA2814882C true CA2814882C (en) | 2019-02-19 |
Family
ID=44800024
Family Applications (1)
| Application Number | Title | Priority Date | Filing Date |
|---|---|---|---|
| CA2814882A Expired - Fee Related CA2814882C (en) | 2010-10-18 | 2011-10-07 | Method and apparatus for determining the relative position between two receivers and use of the apparatus for stabilizing suspended loads |
Country Status (4)
| Country | Link |
|---|---|
| EP (1) | EP2630515B1 (en) |
| CA (1) | CA2814882C (en) |
| DE (1) | DE102011001296A1 (en) |
| WO (1) | WO2012052307A1 (en) |
Families Citing this family (2)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| EP2741108B1 (en) * | 2012-12-07 | 2019-08-28 | Technische Universität München | Method for determining the position of a high orbit satellite |
| US10330790B2 (en) | 2014-05-12 | 2019-06-25 | Linquest Corporation | Multipath rejection using cooperative GPS receivers |
Family Cites Families (5)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| US5991691A (en) * | 1997-02-20 | 1999-11-23 | Raytheon Aircraft Corporation | System and method for determining high accuracy relative position solutions between two moving platforms |
| GB2429128B (en) * | 2005-08-08 | 2009-03-11 | Furuno Electric Co | Apparatus and method for carrier phase-based relative positioning |
| US7702460B2 (en) * | 2006-06-17 | 2010-04-20 | Northrop Grumman Guidance And Electronics Company, Inc. | Estimate of relative position between navigation units |
| EP1972959B1 (en) | 2007-03-22 | 2012-08-15 | DLR Deutsches Zentrum für Luft- und Raumfahrt e.V. | Method for processing a set of signals of a global navigation satellite system with at least three carriers |
| EP2279426B1 (en) | 2008-04-11 | 2012-05-23 | Technische Universität München | Method for vector phase tracking a plurality of global positioning satellite carrier signals |
-
2011
- 2011-03-15 DE DE102011001296A patent/DE102011001296A1/en not_active Withdrawn
- 2011-10-07 WO PCT/EP2011/067554 patent/WO2012052307A1/en active Application Filing
- 2011-10-07 CA CA2814882A patent/CA2814882C/en not_active Expired - Fee Related
- 2011-10-07 EP EP11769845.6A patent/EP2630515B1/en active Active
Also Published As
| Publication number | Publication date |
|---|---|
| WO2012052307A1 (en) | 2012-04-26 |
| CA2814882A1 (en) | 2012-04-26 |
| EP2630515A1 (en) | 2013-08-28 |
| DE102011001296A1 (en) | 2012-04-19 |
| EP2630515B1 (en) | 2019-11-13 |
Similar Documents
| Publication | Publication Date | Title |
|---|---|---|
| CN108508461B (en) | GNSS carrier phase based high-precision positioning integrity monitoring method | |
| US7576690B2 (en) | Position determination with reference data outage | |
| US7982667B2 (en) | Post-processed accuracy prediction for GNSS positioning | |
| US8120527B2 (en) | Satellite differential positioning receiver using multiple base-rover antennas | |
| Reussner et al. | GLONASS inter-frequency biases and their effects on RTK and PPP carrier-phase ambiguity resolution | |
| Giorgi et al. | Carrier phase GNSS attitude determination with the multivariate constrained LAMBDA method | |
| US11125890B2 (en) | Advanced navigation satellite system positioning method and system using seeding information | |
| CN108076662A (en) | The GNSS receiver of the ability of fuzziness is resolved with non-combinatorial formula is used | |
| EP2502091A2 (en) | Detection and correction of anomalous measurements and ambiguity resolution in a global navigation satellite system receiver | |
| EP2479588B1 (en) | Method and apparatus for determining the relative position between two receivers of a satellite navigation system | |
| Banville et al. | Instantaneous cycle‐slip correction for real‐time PPP applications | |
| EP2624013B1 (en) | Method for Determining the Heading of a Moving Object | |
| CN106054223A (en) | Mobile station positioning method, base station and mobile station positioning system | |
| CN108646266A (en) | Construction precisely controlling system and method based on Big Dipper high accuracy positioning | |
| Chen et al. | A geometry-free and ionosphere-free multipath mitigation method for BDS three-frequency ambiguity resolution | |
| EP4103971A1 (en) | System and method for gnss ambiguity resolution | |
| EP4103972A1 (en) | System and method for integer-less gnss positioning | |
| CN109375248A (en) | A kind of Kalman's multimodality fusion location algorithm model and its method serially updated | |
| CA2814882C (en) | Method and apparatus for determining the relative position between two receivers and use of the apparatus for stabilizing suspended loads | |
| US11821999B2 (en) | Attitude determination based on global navigation satellite system information | |
| JP2007163335A (en) | Attitude locating device, attitude locating method, and attitude locating program | |
| WO2017066750A1 (en) | Triple difference formulation for formation flight | |
| RU2624268C1 (en) | Method of determining mutual position of objects by signals of global navigation satellite systems | |
| JP6546730B2 (en) | Satellite signal receiver | |
| Duong et al. | Multi-frequency multi-GNSS PPP: A comparison of two ambiguity resolution methods |
Legal Events
| Date | Code | Title | Description |
|---|---|---|---|
| EEER | Examination request |
Effective date: 20161004 |
|
| MKLA | Lapsed |
Effective date: 20201007 |