Method of determining an alignment error of an antenna and vehicle with an antenna and a detection device

Abstract

A method of determining an alignment error of an antenna is described, wherein the antenna is installed at a vehicle and in cooperation with a detection device, and wherein the detection device is configured to determine a plurality of detections. Determining the plurality of detections comprises emitting a first portion of electromagnetic radiation through the antenna, receiving a second portion of electromagnetic radiation through the antenna, and evaluating the second portion of electromagnetic radiation in dependence of the first portion of electromagnetic radiation in order to localize areas of reflection of the first portion of electromagnetic radiation in the vicinity of the antenna. The method comprises determining a first detection and at least a second detection by using the detection device, and determining the alignment error by means of a joint evaluation of the first detection and the second detection.

Claims

1. A method of determining an alignment error of an antenna, wherein: the antenna is installed on a vehicle and in cooperation with a radar system, the radar system is configured to determine a plurality of detections by: emitting a first portion of electromagnetic radiation through the antenna; receiving a second portion of electromagnetic radiation through the antenna; and evaluating the second portion of electromagnetic radiation in dependence of the first portion of electromagnetic radiation in order to localize areas of reflection of the first portion of electromagnetic radiation in a vicinity of the antenna; and determining the alignment error of the antenna by: determining a first detection and at least a second detection of the plurality of detections by using the radar system, and determining the alignment error by performing a joint evaluation of the first detection and the second detection by at least: evaluating a first algebraic expression (A′, A″) or a second algebraic expression (B′, B″), wherein: the first algebraic expression (A′, A″) is dependent from an angular velocity component (ω.sub.yaw) of the vehicle; the first algebraic expression (A′, A″) is not dependent from a linear velocity component (v.sub.x.sup.host) representing the velocity of the vehicle in a heading direction of the vehicle; the second algebraic expression (B′, B″) is dependent from a linear velocity component (v.sub.x.sup.host) representing the velocity of the vehicle in the heading direction of the vehicle; the second algebraic expression (B′, B″) is not dependent from an angular velocity component (ω.sub.yaw) of the vehicle; and selecting, based on the velocity of the vehicle, one of the first algebraic expression (A′, A″) or the second algebraic expression (B′, B″) for the joint evaluation.

2. The method of claim 1, said method further comprising processing at least one of the first detection and the at least second detection from the radar system in dependence of the alignment error.

3. The method of claim 1, wherein each of the first detection and the second detection comprises an angular position relative to a reference angular position of the antenna.

4. The method of claim 1, wherein each of the first detection and the second detection comprises a velocity of the antenna relative to a velocity of an area of reflection being associated with the first detection and the second detection.

5. The method of claim 1, wherein each of the first detection and the second detection comprises an angular position relative to a reference angular position of the antenna, and each of the first detection and the second detection comprises a velocity of the antenna relative to a velocity of an area of reflection being associated with the first detection and the second detection.

6. The method of claim 1, wherein, from a plurality of detections from the radar system, the joint evaluation is only based on the first detection or the second detection.

7. The method of claim 1, wherein the joint evaluation further comprises one of: evaluating the first algebraic expression (A′, A″) under a condition that the velocity of the vehicle in a heading direction of the vehicle is above a first threshold; or evaluating the first algebraic expression (A′, A″) under a condition that the velocity of the vehicle in a direction being transverse to the heading direction is below a second threshold.

8. The method of claim 7, wherein the first algebraic expression (A′, A″) and the second algebraic expression (B′, B″) do not include a logarithm.

9. The method of claim 1, wherein the joint evaluation further comprises: evaluating the first algebraic expression (A′, A″) under a condition that the velocity of the vehicle in a heading direction of the vehicle is above a first threshold; and evaluating the first algebraic expression (A′, A″) under a condition that the velocity of the vehicle in a direction being transverse to the heading direction is below a second threshold.

10. The method of one of claim 1, wherein the joint evaluation further comprises one of: evaluating the second algebraic expression (B′, B″) under a condition that the velocity of the vehicle in a heading direction of the vehicle is below a third threshold; or evaluating the second algebraic expression (B′, B″) under a condition that the velocity of the vehicle in a direction being transverse to the heading direction is above a fourth threshold.

11. The method of one of claim 1, wherein the joint evaluation further comprises: evaluating the second algebraic expression (B′, B″) under a condition that the velocity of the vehicle in a heading direction of the vehicle is below a third threshold; and evaluating the second algebraic expression (B′, B″) under a condition that the velocity of the vehicle in a direction being transverse to the heading direction is above a fourth threshold.

12. The method of claim 1, wherein the method comprises: determining a plurality of candidates for the alignment error, wherein the alignment error is determined from the candidates by processing the candidates by using at least one of a random sample consensus algorithm or a Kalman filter.

13. The method of claim 1, wherein: the first detection is associated with a first object; the second detection is associated with a second object; and the first object is stationary.

14. The method of claim 1, wherein: the first detection is associated with a first object; the second detection is associated with a second object; the first object and the second object are stationary; and the vehicle is not stationary.

15. A radar system including, a control unit configured to: determine a plurality of detections on the basis of a first portion of electromagnetic radiation emitted through an antenna installed at a vehicle, and a second portion of electromagnetic radiation received through the antenna; and determine an alignment error of the antenna by: determining a first detection and at least a second detection of the plurality of detections; and determining the alignment error by performing of a joint evaluation of the first detection and the second detection including evaluating a first algebraic expression (A′, A″) or a second algebraic expression (B′, B″), wherein: the first algebraic expression (A′, A″) is dependent from an angular velocity component (ω.sub.yaw) of the vehicle; the first algebraic expression (A′, A″) is not dependent from a linear velocity component (v.sub.x.sup.host) representing the velocity of the vehicle in a heading direction of the vehicle; the second algebraic expression (B′, B″) is dependent from a linear velocity component (v.sub.x.sup.host) representing the velocity of the vehicle in the heading direction of the vehicle; and the second algebraic expression (B′, B″) is not dependent from an angular velocity component (ω.sub.yaw) of the vehicle; and selecting, based on the velocity of the vehicle, one of the first algebraic expression (A′, A″) or the second algebraic expression (B′, B″) for performing the joint evaluation.

16. The radar system claim 15, wherein the control unit is configured to perform the joint evaluation by at least one of: evaluating the first algebraic expression (A′, A″) under a condition that the velocity of the vehicle in a heading direction of the vehicle is above a first threshold; evaluating the first algebraic expression (A′, A″) under a condition that the velocity of the vehicle in a direction being transverse to the heading direction is below a second threshold; evaluating the second algebraic expression (B′, B″) under a condition that the velocity of the vehicle in a heading direction of the vehicle is below a third threshold; or evaluating the second algebraic expression (B′, B″) under a condition that the velocity of the vehicle in a direction being transverse to the heading direction is above a fourth threshold.

17. A non-transitory computer-readable storage medium comprising instructions, that when executed, configure at least one processor to determine an alignment error of an antenna of a radar system configured for installation in a vehicle, the alignment error determined by at least: determining a first detection and at least a second detection of a plurality of detections; and determining the alignment error by performing a joint evaluation of the first detection and the second detection including evaluating a first algebraic expression (A′, A″) or a second algebraic expression (B′, B″), wherein: the first algebraic expression (A′, A″) is dependent from an angular velocity component (ω.sub.yaw) of the vehicle; the first algebraic expression (A′, A″) is not dependent from a linear velocity component (v.sub.x.sup.host) representing a velocity of the vehicle in a heading direction of the vehicle; the second algebraic expression (B′, B″) is dependent from a linear velocity component (v.sub.x.sup.host) representing the velocity of the vehicle in the heading direction of the vehicle; the second algebraic expression (B′, B″) is not dependent from an angular velocity component (Ω.sub.yaw) of the vehicle; and performing the joint evaluation by selecting, based on the velocity of the vehicle, one of the first algebraic expression (A′, A″) or the second algebraic expression (B′, B″) for performing the joint evaluation.

18. The non-transitory computer-readable storage medium of claim 7, wherein the instructions, when executed, configure the at least one processor to perform the joint evaluation by at least one of: evaluating the first algebraic expression (A′, A″) under a condition that the velocity of the vehicle in a heading direction of the vehicle is above a first threshold; evaluating the first algebraic expression (A′, A″) under a condition that the velocity of the vehicle in a direction being transverse to the heading direction is below a second threshold; evaluating the second algebraic expression (B′, B″) under a condition that the velocity of the vehicle in a heading direction of the vehicle is below a third threshold; or evaluating the second algebraic expression (B′, B″) under a condition that the velocity of the vehicle in a direction being transverse to the heading direction is above a fourth threshold.

19. The non-transitory computer-readable storage medium of claim 7, wherein the instructions, when executed, further configure a processor to: determine a plurality of candidates for the alignment error; and determine the alignment error from the candidates by processing the candidates using of at least one of a random sample consensus algorithm or a Kalman filter.

Description

BRIEF DESCRIPTION OF DRAWINGS

(1) Further details will now be described, by way of example with reference to the accompanying drawings, in which:

(2) FIG. 1 illustrates a vehicle with two objects in the vicinity of the vehicle; and

(3) FIG. 2 shows an embodiment of a method according to the invention.

DETAILED DESCRIPTION

(4) Reference will now be made in detail to embodiments, examples of which are illustrated in the accompanying drawings. In the following detailed description, numerous specific details are set forth in order to provide a thorough understanding of the various described embodiments. However, it will be apparent to one of ordinary skill in the art that the various described embodiments may be practiced without these specific details. In other instances, well-known methods, procedures, components, circuits, and networks have not been described in detail so as not to unnecessarily obscure aspects of the embodiments.

(5) ‘One or more’ includes a function being performed by one element, a function being performed by more than one element, e.g., in a distributed fashion, several functions being performed by one element, several functions being performed by several elements, or any combination of the above.

(6) It will also be understood that, although the terms first, second, etc. are, in some instances, used herein to describe various elements, these elements should not be limited by these terms. These terms are only used to distinguish one element from another. For example, a first contact could be termed a second contact, and, similarly, a second contact could be termed a first contact, without departing from the scope of the various described embodiments. The first contact and the second contact are both contacts, but they are not the same contact.

(7) The terminology used in the description of the various described embodiments herein is for describing embodiments only and is not intended to be limiting. As used in the description of the various described embodiments and the appended claims, the singular forms “a”, “an” and “the” are intended to include the plural forms as well, unless the context clearly indicates otherwise. It will also be understood that the term “and/or” as used herein refers to and encompasses all possible combinations of one or more of the associated listed items. It will be further understood that the terms “includes,” “including,” “comprises,” and/or “comprising,” when used in this specification, specify the presence of stated features, integers, steps, operations, elements, and/or components, but do not preclude the presence or addition of one or more other features, integers, steps, operations, elements, components, and/or groups thereof.

(8) As used herein, the term “if” is, optionally, construed to mean “when” or “upon” or “in response to determining” or “in response to detecting,” depending on the context. Similarly, the phrase “if it is determined” or “if [a stated condition or event] is detected” is, optionally, construed to mean “upon determining” or “in response to determining” or “upon detecting [the stated condition or event]” or “in response to detecting [the stated condition or event],” depending on the context.

(9) A vehicle 10 comprises four antennas 12a, 12b, 12c, and 12d being installed in the outer corners of the vehicle 10 (indicated very schematically as a rectangle). These installation positions are only exemplary and other distributions and other numbers of antennas are also possible. The antennas 12a, 12b, 12c, 12d are preferably radar antennas (e.g., radomes) which are connected to a radar detection system 02 that uses a control unit 04. The radar detection system 02 is preferably located in the vehicle 10 and can be operated autonomously in the vehicle.

(10) In the following, reference is made to antenna 12a but the corresponding disclosure can be applied mutatis mutandis to the other antennas 12b, 12c, 12d.

(11) Antenna 12a is installed at a predetermined position. Without limitation the predetermined position can be characterized by the orientation of an axis 14. The axis 14 is denoted as the boresight of the antenna 12a. The orientation of axis 14 is expressed relative to an orientation of an x-axis 16 of a predefined coordinate system 18 of the vehicle 10. In FIG. 1, the orientation of axis 14 is expressed by an azimuth angle 20 that is also denoted as θ.sub.boresight or θ.sub.BS.

(12) It is understood that the coordinate system 18 is a Cartesian coordinate system, which also comprises a y-axis 22. The origin 24 of the coordinate system 18 is located in a fixed position in a rear part of the vehicle 10 but other positions are also possible. The x-axis 16 is oriented in a predefined heading direction of the vehicle 10. This is usually the direction in which the driver generally looks through the windscreen of the vehicle 10. It is also the direction of travel of the vehicle 10 if the steering wheel of the vehicle 10 is in a neutral position and the driver performs no steering action. The coordinate system 18 is fixed relative to the vehicle 10.

(13) A first object 26 and a second object 28 are located at different positions in the vicinity of the antenna 12a as shown in FIG. 1. By using the antenna 12a a first detection 30 and a second detection 32 are determined. The detections 30 and 32 represent points of reflection, which are located on the first object 26 and the second object 28, respectively. The first detection 30 comprises a first azimuth angle 34 relative to the axis 14 and the second detection 32 comprises a second azimuth angle 36 relative to the axis 14. The first and second angles 34, 26 are also denoted as angles n and m, respectively.

(14) Under the assumption that the objects 26, 28 are stationary the relative velocity between the antenna 12a and the objects 26, 28 can generally be expressed as:
−v.sub.Dop=v.sub.x.sup.host cos(θ)+v.sub.y.sup.host sin(θ),
wherein v.sub.x.sup.host is the velocity of the antenna 12a in the direction of the x-axis 16 and v.sub.y.sup.host is the velocity of the vehicle 10 in the direction of the y-axis 22. The y-component v.sub.y.sup.host can also be expressed as v.sub.y.sup.host=L.sub.xω.sub.yaw, wherein ω.sub.yaw is the yaw rate of the vehicle 10 and L.sub.x is the distance 38 between the origin 24 and the antenna 12a with respect to the x-axis 16. A virtual axis (not shown), goes through the origin and is oriented orthogonally to the x-axis 16 and the y-axis 22 can also be denoted as the yaw axis of the vehicle 10 (the y-axis 22 may also indicate the rear axis of the vehicle 10).

(15) The relative velocity v.sub.Dop is positive when the vehicle 10 is moving away from the objects 26, 28 and negative when the vehicle 10 is moving towards the objects 26, 28. This is an exemplary definition in line with a definition v.sub.Dop=f.sub.Dλ/2, wherein f.sub.D is the measured Doppler frequency (Doppler shift), and λ is the wavelength of the radar waveform underlying the detections 30, 32, i.e. the wavelength of the emitted radiation.

(16) The angle θ is generally defined as
θ=θ.sub.azi+θ.sub.boresight+θ.sub.align,
wherein θ.sub.azi is one of the angles 34, 36 and θ.sub.align is the alignment error of antenna 12a (not shown). It is understood that the alignment error is not limited to an error of azimuth angle although this is preferred. The alignment error should be zero in order to ensure that the angles 34, 26 can be correctly determined. This is because θ.sub.azi, i.e. the angles 34, 36 are determined with respect to axis 14, wherein the angle 20 of axis 14, i.e. θ.sub.boresight is assumed to match with a predetermined value. A deviation from this value is the alignment error θ.sub.align, as can readily be seen from the above equation.

(17) A ratio A between the detections 30, 32 can be formed as:

(18) $\frac{-v_{\mathrm{Dop}}^n+L_x{\omega }_{\mathrm{yaw}}\sin ({\theta }_{\mathrm{azi}}^n+{\theta }_{\mathrm{boresight}}+{\theta }_{\mathrm{align}})}{-v_{\mathrm{Dop}}^m+L_x{\omega }_{\mathrm{yaw}}\sin ({\theta }_{\mathrm{azi}}^m+{\theta }_{\mathrm{boresight}}+{\theta }_{\mathrm{align}})}=\frac{v_x^{\mathrm{host}}\cos ({\theta }_{\mathrm{azi}}^n+{\theta }_{\mathrm{boresight}}+{\theta }_{\mathrm{align}})}{v_x^{\mathrm{host}}\cos ({\theta }_{\mathrm{azi}}^n+{\theta }_{\mathrm{boresight}}+{\theta }_{\mathrm{align}})}$
wherein the angles θ.sub.azi.sup.m and θ.sub.azi.sup.n correspond to the angles 34 and 36, respectively. The Doppler velocities v.sub.Dop.sup.m and v.sub.Dop.sup.n correspond to the Doppler velocities (or Doppler speeds) determined for the detections 30 and 32, respectively.

(19) As can readily be seen, v.sub.x.sup.host cancels in the above ratio A. The resulting ratio is independent from v.sub.x.sup.host, i.e. independent from the velocity of the vehicle 10 along the x-axis 16. However, the ratio A is dependent from the yaw rate ω.sub.yaw.

(20) A similar ratio B can be formed as:

(21) $\frac{\sin \left({\theta }_{\mathrm{azi}}^n+{\theta }_{\mathrm{boresight}}+{\theta }_{\mathrm{align}}\right)}{\sin \left({\theta }_{\mathrm{azi}}^m+{\theta }_{\mathrm{boresight}}+{\theta }_{\mathrm{align}}\right)}=\frac{v_x^{\mathrm{host}}\cos \left({\theta }_{\mathrm{azi}}^n+{\theta }_{\mathrm{boresight}}+{\theta }_{\mathrm{align}}\right)+v_{\mathrm{Dop}}^n}{v_x^{\mathrm{host}}\cos \left({\theta }_{\mathrm{azi}}^m+{\theta }_{\mathrm{boresight}}+{\theta }_{\mathrm{align}}\right)+v_{\mathrm{Dop}}^m}$
wherein the velocity of the vehicle 10 in the direction of the y-axis 22, i.e. v.sub.y.sup.host=L.sub.xω.sub.yaw cancels. Ratio B is dependent from the velocity v.sub.x.sup.host but independent from the yaw rate ω.sub.yaw. Therefore, ratios A and B are complementing with respect to their dependencies of the linear velocity v.sub.x.sup.host and the yaw rate ω.sub.yaw.

(22) For each of the ratios A and B a solution for the alignment error θ.sub.align can be derived using trigonometric addition theorems and complex numbers, e.g. Euler's formula, known to the skilled person. In this way, ratio A can solved for the alignment error to give a first algebraic expression A′ (see Annex), wherein θ.sub.iso=θ.sub.azi+θ.sub.boresight (with superscripts n and m corresponding to detections 30 and 32) and i is the imaginary unit of complex numbers, commonly interpreted as root of −1. The function e is the exponential function.

(23) In a similar fashion, ratio B can be reshaped and solved to give a second algebraic expression B′ (see Annex) for the alignment error.

(24) It has been found that expressions A′, and B′ can be used to determine the alignment error instead of solutions which are determined by iterative algorithms. The accuracy is the same but using the direct solutions A′ and B′ is computationally more efficient.

(25) Note that expressions A′ and B′ are complex expressions and also require evaluation of a logarithm, i.e. the logarithmic function (log). It is possible to derive real-valued expressions without logarithms by using trigonometric addition theorems and Taylor series expansion, in particular Maclaurin series expansion. Under the assumption that the alignment error is small, the Taylor series expansion can be limited to a low number of terms. In particular, an expansion for the cosine can be limited to the first two terms. In this way, the ratio A can be reformed to an equation, which is quadratic with respect to the alignment error θ.sub.align. A solution of this equation gives an algebraic expression A″ (see Annex).

(26) Similarly, the ratio B can be solved for the alignment error giving a further algebraic expression B″, also shown in the Annex.

(27) Note that expression A″ is dependent from the yaw rate ω.sub.yaw but is not dependent from the velocity v.sub.x.sup.host. In contrast, expression B″ is dependent from the velocity v.sub.x.sup.host but is not dependent from the yaw rate ω.sub.yaw.

(28) In general, each of the expressions A′, A″, B′, and B″ can be used to determine the alignment error. Preferably, the expressions A″ or B″ are used selectively in dependence of the velocity of the vehicle 10. For example if the velocity of the vehicle along the x-axis 16, i.e. the velocity v.sub.x.sup.host is above a first threshold then the expression A″ may be used, which is independent from the velocity v.sub.x.sup.host. A reason for this is that any measurement error of the high velocity v.sub.x.sup.host would have a large effect in the expression A″. Therefore, it is better to rely on the yaw rate ω.sub.yaw, which can be assumed to be small due to the high velocity. In turn, if the yaw rate ω.sub.yaw is high it is better to rely on the low velocity v.sub.x.sup.host. So, in the latter case the expression B″ can be used instead of expression A″.

(29) By selecting different algebraic expressions for determining the alignment error the accuracy of the determined alignment error is usually higher compared to the case that only one expression is used, i.e. the found alignment error better matches the true alignment error.

(30) Note that each of the algebraic expressions A″ and B″ comprises two alternative solutions (the second term is either added to or subtracted from the first term). It is possible to randomly choose one of the alternatives or to choose one of the alternatives by using plausibility checks. For example, it is possible to choose the alternative which better matches with an expected range or distribution of alignment errors due to mechanical reasons of the installation environment of the antenna (e.g., alignment errors above a certain threshold are impossible). The same principles can be applied when using expressions A′ and B′. It is preferred to choose the one of the two alternative solutions that has the lower alignment error.

(31) Each of the expressions A′, A″, B′, and B″ is regarded as a joint evaluation of two detections, namely the first detection 30 and the second detection 32. In this regard, the first and second detections 30, 32 can also be interpreted as a detection pair. The first and second detections 30, 32 can generally be chosen at random from a plurality of detections, e.g. 500 detections. However, predefined criteria can be applied for choosing the first and second detections 30, 32, for example that the detections are located on different objects as shown in FIG. 1.

(32) FIG. 2 shows an overview of the method for determining the alignment error. It starts with block 40, which comprises determining a first and at least a second detection, e.g. the first detection 30 and the second detection 32. In the next block 42 (“joint evaluation”), the detections 30, 32 are evaluated in relation to each other by evaluating at least one of the expressions A′, A″, B′, and B″, preferably only one of the expressions. This gives a value for the alignment error. In a further block 44, a robust processing can be applied to a plurality of solutions for the alignment error. These alignment errors are also denoted as candidates and are all determined through blocks 40, 42, and possibly further processing blocks (not shown). A robust processing algorithm can be applied to the candidates, for example Random Sample Consensus (RANSAC) and/or Kalman Filter. By robust processing a final solution for the alignment error, θ.sub.final, can be determined. This final solution is expected to be more accurate than most of the individual candidates.

(33) On the basis of the final alignment error θ.sub.final any detections, which are acquired by means of the antenna 12a can corrected, i.e. compensated so as to better represent true positions of points of reflections. For example, the alignment error is subtracted from any angle, such as the angles 34, 36 in FIG. 1. In this way, the antenna 12a is electronically aligned and a mechanical alignment is not necessary.

(34) The embodiments of the method as described herein can all be implemented to fulfil real-time capability. This means that all detections which are acquired by using the antenna 12a can readily be compensated with a minimum latency. The antenna 12a can thus be self-aligning with regard to the radar system that is connected with the antenna 12a. The compensated detections have a higher validity than uncompensated (raw) detections. An automated driving application for the vehicle 10, which is based on the compensated detections, is therefore more reliable compared to the case when using uncompensated detections.

(35) While this invention has been described in terms of the preferred embodiments thereof, it is not intended to be so limited, but rather only to the extent set forth in the claims that follow.

(36) Annex

(37) $A^{\prime }\text{:}{\theta }_{\mathrm{align}}=-i\log \left(\frac{\begin{array}{c}{L}_x{\omega }_{\mathrm{yaw}}\left(e^{2{\theta }_{\mathrm{iso}}^{m}i}-e^{2{\theta }_{\mathrm{iso}}^{n}i}\right)i\pm \\ \sqrt{\begin{array}{c}\begin{array}{c}4v_{\mathrm{Dop}}^n{v}_{\mathrm{Dop}}^m\left(e^{{\left({\theta }_{\mathrm{iso}}^n+3{\theta }_{\mathrm{iso}}^m\right)}^i}+e^{{\left(3{\theta }_{\mathrm{iso}}^n+{\theta }_{\mathrm{iso}}^m\right)}^i}\right)-\\ {\left(L_x{\omega }_{\mathrm{yaw}}\right)}^2\end{array}\\ \left(e^{4{\theta }_{{\mathrm{iso}}^i}^m}+e^{4{\theta }_{{\mathrm{iso}}^i}^n}\right)-\left(\begin{array}{c}{\left(2v_{\mathrm{Dop}}^n\right)}^2+{\left(2v_{\mathrm{Dop}}^m\right)}^2-\\ 2{\left(L_x{\omega }_{\mathrm{yaw}}\right)}^2\end{array}\right)e^{2i\left({\theta }_{\mathrm{iso}}^n+{\theta }_{\mathrm{iso}}^m\right)}\end{array}}\end{array}}{2\left(v_{\mathrm{Dop}}^n{e}^{{\left({\theta }_{\mathrm{iso}}^n+2{\theta }_{\mathrm{iso}}^m\right)}^i}-v_{\mathrm{Dop}}^m{e}^{{\left(2{\theta }_{\mathrm{iso}}^n+{\theta }_{\mathrm{iso}}^m\right)}^i}\right)}\right)$$B^{\prime }\text{:}{\theta }_{\mathrm{align}}=-i\log \left(\frac{\begin{array}{c}{v}_x^{\mathrm{host}}\left(e^{2{\theta }_{\mathrm{iso}}^{n}i}-e^{2{\theta }_{\mathrm{iso}}^{m}i}\right)\pm \\ \sqrt{\begin{array}{c}\begin{array}{c}-4v_{\mathrm{Dop}}^n{v}_{\mathrm{Dop}}^m\left(e^{{\left({\theta }_{\mathrm{iso}}^n+3{\theta }_{\mathrm{iso}}^m\right)}^i}+e^{{\left(3{\theta }_{\mathrm{iso}}^n+{\theta }_{\mathrm{iso}}^m\right)}^i}\right)+\\ {\left(v_x^{\mathrm{host}}\right)}^2\end{array}\\ \left(e^{4{\theta }_{{\mathrm{iso}}^i}^m}+e^{4{\theta }_{{\mathrm{iso}}^i}^n}\right)+\left(\begin{array}{c}{\left(2v_{\mathrm{Dop}}^n\right)}^2+{\left(2v_{\mathrm{Dop}}^m\right)}^2-\\ 2{\left(v_x^{\mathrm{host}}\right)}^2\end{array}\right)e^{2i\left({\theta }_{\mathrm{iso}}^n+{\theta }_{\mathrm{iso}}^m\right)}\end{array}}\end{array}}{2\left(v_{\mathrm{Dop}}^n{e}^{{\left({\theta }_{\mathrm{iso}}^n+2{\theta }_{\mathrm{iso}}^m\right)}^i}-v_{\mathrm{Dop}}^m{e}^{{\left(2{\theta }_{\mathrm{iso}}^n+{\theta }_{\mathrm{iso}}^m\right)}^i}\right)}\right)$$B^{″}\text{:}{\theta }_{\mathrm{align}}=\frac{v_{\mathrm{Dop}}^n\cos \left({\theta }_{\mathrm{azi}}^m+{\theta }_{\mathrm{BS}}\right)-v_{\mathrm{Dop}}^m\cos \left({\theta }_{\mathrm{azi}}^m+{\theta }_{\mathrm{BS}}\right)}{v_{\mathrm{Dop}}^n\sin \left({\theta }_{\mathrm{azi}}^m+{\theta }_{\mathrm{BS}}\right)-v_{\mathrm{Dop}}^m\sin \left({\theta }_{\mathrm{azi}}^n+{\theta }_{\mathrm{BS}}\right)}\pm \sqrt{\begin{array}{c}{\left(\frac{v_{\mathrm{Dop}}^n\cos \left({\theta }_{\mathrm{azi}}^m+{\theta }_{\mathrm{BS}}\right)-v_{\mathrm{Dop}}^m\cos \left({\theta }_{\mathrm{azi}}^n+{\theta }_{\mathrm{BS}}\right)}{v_{\mathrm{Dop}}^n\sin \left({\theta }_{\mathrm{azi}}^m+{\theta }_{\mathrm{BS}}\right)-v_{\mathrm{Dop}}^m\sin \left({\theta }_{\mathrm{azi}}^n+{\theta }_{\mathrm{BS}}\right)}\right)}^2+\\ \frac{2\left(v_{\mathrm{Dop}}^n\sin \left({\theta }_{\mathrm{azi}}^m+{\theta }_{\mathrm{BS}}\right)-v_{\mathrm{Dop}}^m\sin \left({\theta }_{\mathrm{azi}}^n+{\theta }_{\mathrm{BS}}\right)-v_x^{\mathrm{host}}\sin \left({\theta }_{\mathrm{azi}}^n-{\theta }_{\mathrm{azi}}^m\right)\right)}{v_{\mathrm{Dop}}^n\sin \left({\theta }_{\mathrm{azi}}^m+{\theta }_{\mathrm{BS}}\right)-v_{\mathrm{Dop}}^m\sin \left({\theta }_{\mathrm{azi}}^n+{\theta }_{\mathrm{BS}}\right)}\end{array}}$$A^{″}\text{:}{\theta }_{\mathrm{align}}=\frac{v_{\mathrm{Dop}}^n\sin \left({\theta }_{\mathrm{azi}}^m+{\theta }_{\mathrm{BS}}\right)-v_{\mathrm{Dop}}^m\sin \left({\theta }_{\mathrm{azi}}^n+{\theta }_{\mathrm{BS}}\right)}{v_{\mathrm{Dop}}^n\cos \left({\theta }_{\mathrm{azi}}^n+{\theta }_{\mathrm{BS}}\right)-v_{\mathrm{Dop}}^m\cos \left({\theta }_{\mathrm{azi}}^m+{\theta }_{\mathrm{BS}}\right)}\pm \sqrt{\begin{array}{c}{\left(\frac{v_{\mathrm{Dop}}^n\sin \left({\theta }_{\mathrm{azi}}^m+{\theta }_{\mathrm{BS}}\right)-v_{\mathrm{Dop}}^m\sin \left({\theta }_{\mathrm{azi}}^n+{\theta }_{\mathrm{BS}}\right)}{v_{\mathrm{Dop}}^m\cos \left({\theta }_{\mathrm{azi}}^n+{\theta }_{\mathrm{BS}}\right)-v_{\mathrm{Dop}}^n\cos \left({\theta }_{\mathrm{azi}}^m+{\theta }_{\mathrm{BS}}\right)}\right)}^2+\\ \frac{2\left(v_{\mathrm{Dop}}^m\cos \left({\theta }_{\mathrm{azi}}^n+{\theta }_{\mathrm{BS}}\right)-v_{\mathrm{Dop}}^n\cos \left({\theta }_{\mathrm{azi}}^m+{\theta }_{\mathrm{BS}}\right)+L_x{\omega }_{\mathrm{yaw}}\sin \left({\theta }_{\mathrm{azi}}^n-{\theta }_{\mathrm{azi}}^m\right)\right)}{v_{\mathrm{Dop}}^m\cos \left({\theta }_{\mathrm{azi}}^n+{\theta }_{\mathrm{BS}}\right)-v_{\mathrm{Dop}}^n\sin \left({\theta }_{\mathrm{azi}}^m+{\theta }_{\mathrm{BS}}\right)}\end{array}}$