CN101975955B - Method for generating universal three-dimensional carrier motion trail in GNSS simulator - Google Patents

Abstract

The invention relates to a method for generating universal three-dimensional carrier motion trail in GNSS simulator, including six steps: firstly, a carrier motion trail user configuration file is pre-generated, and carrier motion segments and trail parameters are set; secondly, when carrier motion trail simulation is started, the preset initial position is taken as origin and a geographic coordinate system is created; thirdly, real time trail simulation calculation is carried out on carrier motion segment in spatial linear motion type; fourthly, real time trail simulation calculation is carried out on carrier motion segment in spatial circle motion type; fifthly, for continuity of motion segments, the last state of a previous motion segment is taken as the initial state of the current motion segment; sixthly, carrier real time position is converted into a geodetic coordinate system to be displayed and output in real time, and the carrier real time position and speed are converted into an ECEF coordinate system, thus supporting further function realization of simulator. The invention can build a universal three-dimensional carrier motion trail generating model and has better practical value and wide application prospect in the technical field of satellite navigation.

Description

Method for generating universal three-dimensional carrier motion trail in a kind of GNSS simulator

(1) technical field

The present invention relates to the movement locus simulation production method of receiver carrier in GPS (Global Position System) (GNSS) satellite signal simulator, relate in particular to the method for generating universal three-dimensional carrier motion trail in a kind of GNSS simulator, belong to the Satellite Navigation Technique field.

(2) background technology

The GNSS satellite signal simulator is by producing the movement locus of satellite signal receiver carrier, calculate in real time the position of receiver, and judge satellite visibility in conjunction with the real time position of satellite, the receiver real-time monitored amounts such as Calculation of Satellite signal power, pseudorange, Doppler, carrier phase, realize the spreading code of navigation data is modulated and carrier modulation with this, simulation produces the satellite navigation signals that receiver receives in the practical application scene, becomes the dummy source in satellite signal receiver development and the testing authentication.

Compare and directly utilize the real satellite signal, the GNSS satellite signal simulator can provide the test condition of accurate controlled, reproducible simulated environment and abnormal condition, and the efficiency of research and development of receiver is guaranteed.Along with GPS System in USA modernization and Russian GLONASS system improve upgrading, and the construction of European Galileo system and China's Beidou satellite navigation system (above-mentioned GPS (Global Position System) can be generally called the GNSS system), the GNSS simulator continues to be subject to the concern of industry and military service.

The external GNSS satellite signal simulator of having developed Multiple Type, as, the GNSS simulator of the GSS series analog device of the GPS simulator of U.S. CAST company, Britain Spirent company, German IFEN company, but mostly function is fixed, and navigational satellite system, signal frequency point and structure, the carrier dynamic dispatching that can simulate are strictly limited.

Analog subscriber dynamically is one of important step of GNSS simulator, accurately simulates the motor pattern of receiver carrier, is conducive to the dynamic property of receiver is carried out Measurement and evaluation; And the simulation of dynamic navigation signal produces the capturing and tracking performances test can support receiver, and the analysis and evaluation of the performance such as bearing accuracy.The simulation of receiver carrier movement track can be based on the built-in carrier movement model of signal simulator, form that also can the actual measurement track data file is written into the GNSS signal simulator, but the implementation of built-in carrier movement model is more flexible, accuracy and repeatable aspect more outstanding.

(3) summary of the invention

1, purpose

In existing GNSS simulator product, relevant receiver carrier movement track generates, usually at first divide some bearer types, under bearer type, divide again the motor pattern of distinguishing with the carrier movement characteristic, and it is self-defined to support that under motor pattern the user carries out the movement locus parameter related with motor pattern, the above-mentioned flow process of foundation is finally finished the definition of carrier track, and simulator generates the carrier movement track accordingly.Take Spirent GSS series analog device as example, the user option of its bearer type is divided into: simple motion (only comprising planar circumferential motion and the motion of plane rectangular path), vehicle, naval vessel, aircraft.Under the option of above-mentioned bearer type, the user need to carry out the motor pattern related with bearer type and select, take the aircraft bearer type as example, according to the kinetic characteristic of different mission phases, its motor pattern is divided into: climb, accelerate straight line, at the uniform velocity straight line, at the uniform velocity turn, accelerate to turn, spiral etc.So that at the uniform velocity the turning motion pattern is as example, the definable movement locus parameter of user comprises: variable quantity, the centripetal acceleration at vector angle in the motor segment; To accelerate the rectilinear motion pattern as example, the definable movement locus parameter of user comprises: the duration of accelerating sections or lasting distance, acceleration or end speed.

Extendability is the subject matter of above-mentioned simulator receiver carrier movement track generation technique, and namely bearer type, motor pattern, definable trajectory parameters are subject to loaded down with trivial details and specific restriction.Receiver test and test by the simulator support, usually just utilize actual GNSS signal to test committed step before, thereby become one of characteristics of general GNSS simulator towards specific application scenarios and typical testing experiment, this is the main cause that the summary locus model was fixed and adopted to GNSS simulator carrier dynamic mode.For the GNSS system in construction still (as, the Galileo system, Beidou satellite navigation system), generally utilize actual satellite-signal and infeasible, because the impact in system Construction cycle, the actual signal test may far lag behind the simulator test, thereby corresponding GNSS simulator is had higher requirement, and more comprehensively testing experiment needs the carrier movement trace simulation of simulator to possess extended capability, possesses the ability that reflects various dynamic versatilities, possesses the Simulation of Complex track before system builds up and possesses service ability.

The objective of the invention is in order to overcome the limitation of existing GNSS simulator carrier movement track generation technique, method for generating universal three-dimensional carrier motion trail in a kind of GNSS simulator is provided, the combination that it can be decomposed into the carrier movement track of any complex form space line motor segment and space circumference motor segment is similar to, by the mutual conversion of attitude description and a plurality of coordinate systems, set up general three-dimensional carrier movement locus production model.

2, technical scheme

Purpose of the present invention is achieved through the following technical solutions: the method for generating universal three-dimensional carrier motion trail in a kind of GNSS simulator, and the method concrete steps are as follows:

Step 1: generate in advance carrier movement track user profile, configuration file has defined carrier movement segmentation and trajectory parameters setting;

The carrier initial position is set to the coordinate (λ in the earth coordinates ₀,

h ₀), i.e. longitude, latitude and elevation.The carrier initial velocity be set in the geographic coordinate system east orientation, north orientation and day to initial velocity (v _{E, 0}, v _{N, 0}, v _{U, 0});

About the trajectory parameters setting of carrier movement segmentation, for the rectilinear motion type, the trajectory parameters that needs to arrange comprises: the rectilinear motion type identification, and the motor segment duration, the east orientation acceleration, the north orientation acceleration, the sky is to acceleration; For the circular motion type, the trajectory parameters that needs to arrange comprises: circular motion type identification, motor segment duration, direction of motion (moving clockwise or counterclockwise), radius of a circle, linear acceleration, the deflection of the relative geographic coordinate system of circular motion in-plane.

Step 2: when the carrier movement trace simulation begins, set up geographic coordinate system, initial position (x take the initial position (being longitude, latitude and elevation) that arranges as initial point ₀, y ₀, z ₀) be (0,0,0);

The x axle of geographic coordinate system (claiming again the e axle) points to east along local parallel tangent line, and y axle (claiming again the n axle) is along local warp tangent line energized north, and z axle (claiming again the u axle) points to zenith along local geographic vertical (earth reference ellipsoid surface normal).For the carrier movement trace simulation of dynamic range in 180km, the geographic coordinate system of this initial setting up can be considered three axis directions and remains unchanged, and caused error can be ignored, and then initial geographic coordinate system will be held in this distance range.Before the carrier movement trace simulation finishes, if carrier dynamically exceeds this distance range, then need to upgrade geographic coordinate system;

Step 3: obtain the trajectory parameters of carrier movement the first segmentation, if be the space line type of sports, in geographic coordinate system, according to initial velocity (v _{E, 0}, v _{N, 0}, v _{U, 0}), acceleration (a _e, a _n, a _u) setting, can determine present position, the speed of carrier;

Motor segment is in the duration, according to t _K-1Position (the x of emulation moment epoch carrier _K-1, y _K-1, z _K-1), speed (v _{E, k-1}, v _{N, k-1}, v _{U, k-1}) calculate t _kMoment carrier positions (x _k, y _k, z _k), speed (v _{E, k}, v _{N, k}, v _{U, k}) method as follows:

(a) computing time interval of delta t

Δt＝t _k-t _k-1

(b) calculate t _K-1The time be carved into t _kChange of distance amount (Δ s constantly _e, Δ s _n, Δ s _u)

$\left\{\begin{array}{c}\mathrm{\Δ}{s}_e=v_{e,k-1}\·\mathrm{\Δt}+\frac{1}{2}{a}_e\·\mathrm{\Δ}{t}^2\\ \mathrm{\Δ}{s}_n=v_{n,k-1}\·\mathrm{\Δt}+\frac{1}{2}{a}_n\·\mathrm{\Δ}{t}^2\\ \mathrm{\Δ}{s}_u=v_{u,k-1}\·\mathrm{\Δt}+\frac{1}{2}{a}_u\·\mathrm{\Δ}{t}^2\end{array}\right.$

(c) calculate t _kMoment bearer rate (v _{E, k}, v _{N, k}, v _{U, k})

$\left\{\begin{array}{c}{v}_{e,k}=v_{e,k-1}+a_e\·\mathrm{\Δt}\\ v_{n,k}=v_{n,k-1}+a_n\·\mathrm{\Δt}\\ v_{u,k}=v_{u,k-1}+a_u\·\mathrm{\Δt}\end{array}\right.$

(d) calculate t _kMoment carrier positions (x _k, y _k, z _k)

$\left\{\begin{array}{c}{x}_k=x_{k-1}+\mathrm{\Δ}{s}_e\\ y_k=y_{k-1}+\mathrm{\Δ}{s}_n\\ z_k=z_{k-1}+\mathrm{\Δ}{s}_u\end{array}\right.$

Step 4: if carrier movement first is segmented into space circumference type of sports, (get the angle of plane of movement direction and geographic coordinate system z axle according to the relative geographic coordinate system deflection of plane of movement γ, take z axle right side as positive-angle) setting, at first determine the two dimensional surface of circular motion in the geographic coordinate system, the x axle on this plane is geographic coordinate system x axle (e axle), and the y axle is vertical with the x axle;

Secondly, according to initial velocity (v _{E, 0}, v _{N, 0}, v _{U, 0}), the setting of radius of a circle R and linear acceleration a, can determine present position, the speed of carrier in the circular motion plane.Motor segment calculates t in the duration _kCarrier positions in the emulation moment epoch circular motion plane

Speed (v _{X, k}, v _{Y, k}) method as follows:

(a) computing time interval of delta t

Δt＝t _k-t _k-1

(b) calculate t _K-1The time be carved into t _kCorner variation delta φ constantly

$\mathrm{\Δ\φ}=-\mathrm{sign}\·(\frac{\left|v_{k-1}\right|}{R}\·\mathrm{\Δt}+\frac{1}{2}\·\frac{a}{R}\·\mathrm{\Δ}{t}^2)$

Wherein, | .| is absolute value operators,

And $v_{k-1}=\sqrt{v_{e,k-1}^2+v_{n,k-1}^2+v_{u,k-1}^2}$

(c) calculate t _kCarrier position angle φ in the moment plane of movement _k

φ _k＝φ _k-1+Δφ

And initial orientation ${\mathrm{\&phi;}}_{k-1}=\left\{\begin{array}{cc}\arctan \frac{v_{x,k-1}}{v_{y,k-1}},& \frac{v_{x,k-1}}{v_{y,k-1}}>0\\ \mathrm{\&pi;}-\arctan \frac{v_{x,k-1}}{v_{y,k-1}},& \frac{v_{x,k-1}}{v_{y,k-1}}<0\end{array}\right.$

(d) calculate t _kBearer rate (v in the moment plane of movement _{X, k}, v _{Y, k})

$\left\{\begin{array}{c}{v}_{x,k}=\mathrm{sign}\&CenterDot;\left|v_k\right|\&CenterDot;\sin {\mathrm{\&phi;}}_k\\ v_{y,k}=-\mathrm{sign}\&CenterDot;\left|v_k\right|\&CenterDot;\cos {\mathrm{\&phi;}}_k\end{array}\right.$

And | v _k|=| v _K-1|+a Δ t

(e) calculate t _kCarrier positions in the moment plane of movement

$\left\{\begin{array}{c}{x}_k^p=\mathrm{sign}\&CenterDot;R\cos {\mathrm{\&phi;}}_k+\mathrm{sign}\&CenterDot;\frac{v_{y0}\&CenterDot;R}{\sqrt{v_{x0}^2+v_{y0}^2}}+x_0^p\\ y_k^p=\mathrm{sign}\&CenterDot;R\sin {\mathrm{\&phi;}}_k-\mathrm{sign}\&CenterDot;\frac{v_{x0}\&CenterDot;R}{\sqrt{v_{x0}^2+v_{y0}^2}}+y_0^p\end{array}\right.$

And $\left[\begin{array}{c}{x}_0^p\\ y_0^p\end{array}\right]=\left[\begin{array}{ccc}1& 0& 0\\ 0& \cos \mathrm{\&gamma;}& -\sin \mathrm{\&lambda;}\end{array}\right]\&CenterDot;\left[\begin{array}{c}{x}_0\\ y_0\\ z_0\end{array}\right]$

At last, with the position of carrier in the plane of movement and the rate conversion corresponding sports state parameter in the geographic coordinate system, conversion method is as follows:

(a) determine that plane of movement coordinate system p is to the rotation of coordinate matrix of geographic coordinate system L

$R_p^L=\left[\begin{array}{cc}1& 0\\ 0& \cos \mathrm{\&gamma;}\\ 0& -\sin \mathrm{\&gamma;}\end{array}\right]$

(b) calculate t _kMoment geographic coordinate system bearer rate (v _{E, k}, v _{N, k}, v _{U, k})

$\left[\begin{array}{c}{v}_{e,k}\\ v_{n,k}\\ v_{u,k}\end{array}\right]=R_p^L\&CenterDot;\left[\begin{array}{c}{v}_{x,k}\\ v_{y,k}\end{array}\right]$

(c) calculate t _kMoment geographic coordinate system carrier positions (x _k, y _k, z _k)

$\left[\begin{array}{c}{x}_k\\ y_k\\ z_k\end{array}\right]=R_p^L\&CenterDot;\left[\begin{array}{c}{x}_k^p\\ y_k^p\end{array}\right]$

Step 5: the user according to straight line or circular motion type arranges, and according to step 3, step 4, processes successively the subsequent motion segmentation.For the linking between the motion segmentation, more than the last current state of a motor segment be the initial equilibrium state of current motor segment, namely go up the carrier end position (x of a motor segment _m, y _m, z _m), end speed (v _{E, m}, v _{N, m}, v _{U, m}) become respectively the initial position (x of current motor segment ₀, y ₀, z ₀), initial velocity (v _{E, 0}, v _{N, 0}, v _{U, 0}), namely

$\left[\begin{array}{c}{x}_m\\ y_m\\ z_m\end{array}\right]=\left[\begin{array}{c}{x}_0\\ y_0\\ z_0\end{array}\right],\left[\begin{array}{c}{v}_{e,m}\\ v_{n,m}\\ v_{u,m}\end{array}\right]=\left[\begin{array}{c}{v}_{e,0}\\ v_{n,0}\\ v_{u,0}\end{array}\right]$

Step 6: each emulation epoch at simulator the carrier present position in the geographic coordinate system is transformed into earth coordinates, is used for user interface and shows in real time output; And carrier present position, rate conversion judged with further support satellite visibility to earth rectangular coordinate system (claiming again the ECEF coordinate system) the simulator functions such as signal condition calculation of parameter such as pseudorange, Doppler realize;

(a) calculate t _kCorresponding carrier positions coordinate (λ in the moment earth coordinates _k,

h _k)

Wherein,

A is earth reference ellipsoid semi-major axis, and e is earth reference ellipsoid excentricity.

Be convenient to for the purpose of the numerical evaluation, consider R _NAnd R _MApproximate data, namely

And ellipticity $f=1-\sqrt{1-e^2}.$

(b) calculate t _kCorresponding carrier positions coordinate (x in the moment ECEF coordinate system _{E, k}, y _{E, k}, z _{E, k})

Wherein, λ,

Be carrier positions coordinate (x _k, y _k, z _k) corresponding warp, latitude, and (x _Oe, y _Oe, z _Oe) be geographic coordinate system initial point (λ ₀,

h ₀) respective coordinates in the ECEF coordinate system, can calculate by following formula:

And

(c) calculate t _kCorresponding bearer rate (v in the moment ECEF coordinate system _{X, k}, v _{Y, k}, v _{Z, k})

3, advantage and effect

Can be found out by technique scheme, compare with domestic and international existing GNSS simulator carrier movement orbit generation method that the carrier orbit generation method that the present invention proposes has following technological merit:

(1) the carrier movement trace simulation has versatility, does not distinguish bearer type, thinks that the behavioral characteristics of carrier track self namely reflects bearer type; The motor pattern of containing the satellite signal receiver carrier that is applicable to the GNSS simulator with space line motion, two kinds of motor patterns of space circular motion.

(2) the based on motion trajectory segment decomposes approximate and introduces attitude and describe, analog capability with the complex carrier movement locus that is applicable to the GNSS simulator, the degree of approximation to complicated track only depends on the fine degree that motor segment is divided, and the small scale of the dynamic details of supporting carrier is accurately described.

(3) adopt carrier movement track user profile mode, the flexible configuration of carrier trajectory parameters, convenient supports the user to produce by complete self-defining carrier movement trace simulation, makes the carrier trace simulation have extendability.

(4) description of drawings

Fig. 1 is method for generating universal three-dimensional carrier motion trail FB(flow block) of the present invention

Fig. 2 is carrier movement track user profile form synoptic diagram of the present invention

Fig. 3 is that carrier movement track of the present invention generates exemplary plot

Symbol description is as follows among the figure:

+ e: east orientation+n: north orientation+u: day to

(5) embodiment

A kind of method for generating universal three-dimensional carrier motion trail that is applicable to the GNSS simulator of the present invention, described method flow block diagram as shown in Figure 1.The method is by the software desk Implementation of GNSS simulator, before the method implementation, human-computer interaction interface based on the GNSS simulator, select navigational system and the frequency of required emulation, emulation zero-time and simulation time length are set, choose the ephemeris source file, it is dynamic that the receiver carrier is set, select whether to introduce the various error terms that will convert as range error, etc.When the receiver carrier being set when dynamic, needing to select carrier movement track user profile, thereby enter the flow process of the method for the invention.

The method specifically realizes by following steps:

Step 1: generate in advance carrier movement track user profile, configuration file has defined carrier movement segmentation and trajectory parameters setting.The format description of relevant user profile, as shown in Figure 2.

The carrier initial position is set to the coordinate (λ in the earth coordinates ₀,

h ₀), i.e. longitude, latitude and elevation.The carrier initial velocity be set in the geographic coordinate system east orientation, north orientation and day to initial velocity (v _{E, 0}, v _{N, 0}, v _{U, 0}).

About the trajectory parameters setting of carrier movement segmentation, for the rectilinear motion type, the trajectory parameters that needs to arrange comprises: the rectilinear motion type identification, and the motor segment duration, the east orientation acceleration, the north orientation acceleration, the sky is to acceleration; For the circular motion type, the trajectory parameters that needs to arrange comprises: circular motion type identification, motor segment duration, direction of motion (moving clockwise or counterclockwise), radius of a circle, linear acceleration, the deflection of the relative geographic coordinate system of circular motion in-plane.

Step 2: when the carrier movement trace simulation begins, set up geographic coordinate system, initial position (x take the initial position (being longitude, latitude and elevation) that arranges as initial point ₀, y ₀, z ₀) be (0,0,0).

The x axle of geographic coordinate system (claiming again the e axle) points to east along local parallel tangent line, and y axle (claiming again the n axle) is along local warp tangent line energized north, and z axle (claiming again the u axle) points to zenith along local geographic vertical (earth reference ellipsoid surface normal).For the carrier movement trace simulation of dynamic range in 180km, the geographic coordinate system of this initial setting up can be considered three axis directions and remains unchanged, and caused error can be ignored, and then initial geographic coordinate system will be held in this distance range.Before the carrier movement trace simulation finishes, if carrier dynamically exceeds this distance range, then need to upgrade geographic coordinate system.

Step 3: obtain the trajectory parameters of carrier movement the first segmentation, if be the space line type of sports, in geographic coordinate system, according to initial velocity (v _{E, 0}, v _{N, 0}, v _{U, 0}), acceleration (a _e, a _n, a _u) setting, can determine present position, the speed of carrier.

Motor segment is in the duration, according to t _K-1Position (the x of emulation moment epoch carrier _K-1, y _K-1, z _K-1), speed (v _{E, k-1}, v _{N, k-1}, v _{U, k-1}) calculate t _kMoment carrier positions (x _k, y _k, z _k), speed (v _{E, k}, v _{N, k}, v _{U, k}) method as follows:

(a) computing time interval of delta t

Δt＝t _k-t _k-1

(b) calculate t _K-1The time be carved into t _kChange of distance amount (Δ s constantly _e, Δ s _n, Δ s _u)

$\left\{\begin{array}{c}\mathrm{\&Delta;}{s}_e=v_{e,k-1}\&CenterDot;\mathrm{\&Delta;t}+\frac{1}{2}{a}_e\&CenterDot;\mathrm{\&Delta;}{t}^2\\ \mathrm{\&Delta;}{s}_n=v_{n,k-1}\&CenterDot;\mathrm{\&Delta;t}+\frac{1}{2}{a}_n\&CenterDot;\mathrm{\&Delta;}{t}^2\\ \mathrm{\&Delta;}{s}_u=v_{u,k-1}\&CenterDot;\mathrm{\&Delta;t}+\frac{1}{2}{a}_u\&CenterDot;\mathrm{\&Delta;}{t}^2\end{array}\right.$

(c) calculate t _kMoment bearer rate (v _{E, k}, v _{N, k}, v _{U, k})

$\left\{\begin{array}{c}{v}_{e,k}=v_{e,k-1}+a_e\&CenterDot;\mathrm{\&Delta;t}\\ v_{n,k}=v_{n,k-1}+a_n\&CenterDot;\mathrm{\&Delta;t}\\ v_{u,k}=v_{u,k-1}+a_u\&CenterDot;\mathrm{\&Delta;t}\end{array}\right.$

(d) calculate t _kMoment carrier positions (x _k, y _k, z _k)

$\left\{\begin{array}{c}{x}_k=x_{k-1}+\mathrm{\&Delta;}{s}_e\\ y_k=y_{k-1}+\mathrm{\&Delta;}{s}_n\\ z_k=z_{k-1}+\mathrm{\&Delta;}{s}_u\end{array}\right.$

Step 4: if carrier movement first is segmented into space circumference type of sports, (get the angle of plane of movement direction and geographic coordinate system z axle according to the relative geographic coordinate system deflection of plane of movement γ, take z axle right side as positive-angle) setting, at first determine the two dimensional surface of circular motion in the geographic coordinate system, the x axle on this plane is geographic coordinate system x axle (e axle), and the y axle is vertical with the x axle.

Secondly, according to initial velocity (v _{E, 0}, v _{N, 0}, v _{U, 0}), the setting of radius of a circle R and linear acceleration a, can determine present position, the speed of carrier in the circular motion plane.Motor segment calculates t in the duration _kCarrier positions in the emulation moment epoch circular motion plane

Speed (v _{X, k}, v _{Y, k}) method as follows:

(a) computing time interval of delta t

Δt＝t _k-t _k-1

(b) calculate t _K-1The time be carved into t _kCorner variation delta φ constantly

$\mathrm{\&Delta;\&phi;}=-\mathrm{sign}\&CenterDot;(\frac{\left|v_{k-1}\right|}{R}\&CenterDot;\mathrm{\&Delta;t}+\frac{1}{2}\&CenterDot;\frac{a}{R}\&CenterDot;\mathrm{\&Delta;}{t}^2)$

Wherein, | .| is absolute value operators,

And $v_{k-1}=\sqrt{v_{e,k-1}^2+v_{n,k-1}^2+v_{u,k-1}^2}$

(c) calculate t _kCarrier position angle φ in the moment plane of movement _k

φ _k＝φ _k-1+Δφ

And initial orientation ${\mathrm{\&phi;}}_{k-1}=\left\{\begin{array}{cc}\arctan \frac{v_{x,k-1}}{v_{y,k-1}},& \frac{v_{x,k-1}}{v_{y,k-1}}>0\\ \mathrm{\&pi;}-\arctan \frac{v_{x,k-1}}{v_{y,k-1}},& \frac{v_{x,k-1}}{v_{y,k-1}}<0\end{array}\right.$

(d) calculate t _kBearer rate (v in the moment plane of movement _{X, k}, v _{Y, k})

$\left\{\begin{array}{c}{v}_{x,k}=\mathrm{sign}\&CenterDot;\left|v_k\right|\&CenterDot;\sin {\mathrm{\&phi;}}_k\\ v_{y,k}=-\mathrm{sign}\&CenterDot;\left|v_k\right|\&CenterDot;\cos {\mathrm{\&phi;}}_k\end{array}\right.$

And | v _k|=| v _K-1|+a Δ t

(e) calculate t _kCarrier positions in the moment plane of movement

$\left\{\begin{array}{c}{x}_k^p=\mathrm{sign}\&CenterDot;R\cos {\mathrm{\&phi;}}_k+\mathrm{sign}\&CenterDot;\frac{v_{y0}\&CenterDot;R}{\sqrt{v_{x0}^2+v_{y0}^2}}+x_0^p\\ y_k^p=\mathrm{sign}\&CenterDot;R\sin {\mathrm{\&phi;}}_k-\mathrm{sign}\&CenterDot;\frac{v_{x0}\&CenterDot;R}{\sqrt{v_{x0}^2+v_{y0}^2}}+y_0^p\end{array}\right.$

And $\left[\begin{array}{c}{x}_0^p\\ y_0^p\end{array}\right]=\left[\begin{array}{ccc}1& 0& 0\\ 0& \cos \mathrm{\&gamma;}& -\sin \mathrm{\&lambda;}\end{array}\right]\&CenterDot;\left[\begin{array}{c}{x}_0\\ y_0\\ z_0\end{array}\right]$

At last, with the position of carrier in the plane of movement and the rate conversion corresponding sports state parameter in the geographic coordinate system, conversion method is as follows:

(a) determine that plane of movement coordinate system p is to the rotation of coordinate matrix of geographic coordinate system L

$R_p^L=\left[\begin{array}{cc}1& 0\\ 0& \cos \mathrm{\&gamma;}\\ 0& -\sin \mathrm{\&gamma;}\end{array}\right]$

(b) calculate t _kMoment geographic coordinate system bearer rate (v _{E, k}, v _{N, k}, v _{U, k})

$\left[\begin{array}{c}{v}_{e,k}\\ v_{n,k}\\ v_{u,k}\end{array}\right]=R_p^L\&CenterDot;\left[\begin{array}{c}{v}_{x,k}\\ v_{y,k}\end{array}\right]$

(c) calculate t _kMoment geographic coordinate system carrier positions (x _k, y _k, z _k)

$\left[\begin{array}{c}{x}_k\\ y_k\\ z_k\end{array}\right]=R_p^L\&CenterDot;\left[\begin{array}{c}{x}_k^p\\ y_k^p\end{array}\right]$

Step 5: the user according to straight line or circular motion type arranges, and according to step 3, step 4, processes successively the subsequent motion segmentation.For the linking between the motion segmentation, more than the last current state of a motor segment be the initial equilibrium state of current motor segment, namely go up the carrier end position (x of a motor segment _m, y _m, z _m), end speed (v _{E, m}, v _{N, m}, v _{U, m}) become respectively the initial position (x of current motor segment ₀, y ₀, z ₀), initial velocity (v _{E, 0}, v _{N, 0}, v _{U, 0}), namely

$\left[\begin{array}{c}{x}_m\\ y_m\\ z_m\end{array}\right]=\left[\begin{array}{c}{x}_0\\ y_0\\ z_0\end{array}\right],\left[\begin{array}{c}{v}_{e,m}\\ v_{n,m}\\ v_{u,m}\end{array}\right]=\left[\begin{array}{c}{v}_{e,0}\\ v_{n,0}\\ v_{u,0}\end{array}\right]$

Step 6: each emulation epoch at simulator the carrier present position in the geographic coordinate system is transformed into earth coordinates, is used for user interface and shows in real time output; And carrier present position, rate conversion judged with further support satellite visibility to earth rectangular coordinate system (claiming again the ECEF coordinate system) the simulator functions such as signal condition calculation of parameter such as pseudorange, Doppler realize.

(a) calculate t _kCorresponding carrier positions coordinate (λ in the moment earth coordinates _k,

h _k)

Wherein, A is earth reference ellipsoid semi-major axis, and e is earth reference ellipsoid excentricity.

Be convenient to for the purpose of the numerical evaluation, consider R _NAnd R _MApproximate data, namely

And ellipticity $f=1-\sqrt{1-e^2}.$

(b) calculate t _kCorresponding carrier positions coordinate (x in the moment ECEF coordinate system _{E, k}, y _{E, k}, z _{E, k})

Wherein, λ,

Be carrier positions coordinate (x _k, y _k, z _k) corresponding warp, latitude, and (x _Oe, y _Oe, z _Oe) be geographic coordinate system initial point (λ ₀,

h ₀) corresponding coordinate in the ECEF coordinate system, can calculate by following formula:

And

(c) calculate t _kCorresponding bearer rate (v in the moment ECEF coordinate system _{X, k}, v _{Y, k}, v _{Z, k})

Embodiment

The method for generating universal three-dimensional carrier motion trail that provides provided by the invention is provided, take simulation generation carrier movement track as shown in Figure 3 as example, and with WGS-84 terrestrial coordinate system example, the specific implementation step is described.

Step 1: GNSS simulator user arranges the carrier movement trajectory parameters, generates carrier movement track user profile.

For example, the carrier initial position is set: 116 ° of longitudes, 39 ° in latitude, elevation 100m; Carrier initial velocity in the geographic coordinate system is set: east orientation 100m/s, north orientation 0m/s, the sky is to 0m/s; Carrier trajectory segment and parameter are set, and this track is comprised of 10 rectilinear motion segmentations and circular motion segmentation, and former 5 are segmented into the example explanation, are respectively:

(a) first be segmented into rectilinear motion, east orientation, north orientation and sky are 0 to acceleration, last 5s;

(b) second be segmented into circular motion, the deflection of the relative geographic coordinate system of plane of movement is 0, and radius of a circle is 200m, clockwise direction of motion, and linear acceleration is 0, lasts 20s;

(c) the 3rd be segmented into rectilinear motion, east orientation, north orientation and sky are 0 to acceleration, last 5s;

(d) the 4th be segmented into circular motion, the deflection of the relative geographic coordinate system of plane of movement is 45 °, and radius of a circle is 200m, clockwise direction of motion, and linear acceleration is 40m/s ², last 5s;

(e) the 5th be segmented into rectilinear motion, the east orientation acceleration is 65m/s ², the north orientation acceleration is 34m/s ², day to acceleration be-40m/s ², last 5s.

Step 2: read carrier movement track user profile, set up geographic coordinate system take the initial position that arranges as initial point.The x axle of this geographic coordinate system (claiming again the e axle) points to east along 39 ° of parallel tangent lines, and y axle (claiming again the n axle) is along 116 ° of warp tangent line energized north, z axle (claiming again the u axle) and x and the vertical zenith that points in the determined plane of y axle.

Step 3: real time modelling carrier track, calculating carrier instantaneous velocity and position coordinates are set according to the motion segmentation trajectory parameters in the step 1.

For the first segmentation rectilinear motion, carrier is from aforementioned geographic coordinate system initial point (0,0,0) initial, with 100m/s initial velocity east orientation linear uniform motion, last 5s (being Δ t=5s), then in the finish time of this motor segment, change of distance amount, speed and the position coordinates of carrier are respectively:

$\left\{\begin{array}{c}\mathrm{\&Delta;}{s}_e=500\\ \mathrm{\&Delta;}{s}_n=0\\ \mathrm{\&Delta;}{s}_u=0\end{array}\right.\left(m\right),$ $\left\{\begin{array}{c}{v}_{e,m}=100\\ v_{n,m}=0\\ v_{u,m}=0\end{array}\right.(m/s),$ $\left\{\begin{array}{c}{x}_m=500\\ y_m=0\\ z_m=0\end{array}\right.\left(m\right)$

Step 4: for the second segmentation circular motion, carrier is from the last position (500 of a upper motor segment, 0,0) initial, more than the end speed rate of a motor segment be the clockwise uniform circular motion of first speed, plane of movement maintains in the surface level of aforementioned geographic coordinate system (being γ=0), lasts 20s (being Δ t=20s), then at the initial time of this motor segment, the initial orientation of carrier, first speed, initial position are respectively:

φ ₀＝0.5πrad， $v_0=100m/s,\left\{\begin{array}{c}{x}_0^p=500\\ y_0^p=0\end{array}\right.\left(m\right)$

And in the finish time of this motor segment, the corner variable quantity of carrier, position angle, plane of movement speed and plane of movement coordinate are respectively:

Δφ＝-10rad，φ _m＝-8.4292rad， $\left\{\begin{array}{c}{v}_{x,m}=-83.9072\\ v_{y,m}=54.4021\end{array}\right.(m/s),$ $\left\{\begin{array}{c}{x}_m^p=391.1958\\ y_m^p=-367.8143\end{array}\right.\left(m\right)$

Further realize the conversion to geographic coordinate system, then rotation of coordinate matrix, geographic coordinate system bearer rate and position coordinates are respectively:

$R_p^L=\left[\begin{array}{cc}1& 0\\ 0& 1\\ 0& 0\end{array}\right],\left\{\begin{array}{c}{v}_{e,m}=-83.9072\\ v_{n,.m}=54.4021\\ v_{u,m}=0\end{array}\right.(m/s),$ $\left\{\begin{array}{c}{x}_m=391.1958\\ y_m=-367.8143\\ z_m=0\end{array}\right.\left(m\right)$

Step 5: according to step 3, step 4, process successively the subsequent motion segmentation.

(a) for the 3rd segmentation, carrier is from the last position (391.1958 of a upper motor segment,-367.8143,0) initial, with (83.9027,54.4021,0) initial velocity linear uniform motion, last 5s (being Δ t=5s), then in the finish time of this motor segment, change of distance amount, speed and the position coordinates of carrier are respectively:

$\left\{\begin{array}{c}\mathrm{\&Delta;}{s}_e=-419.5358\\ \mathrm{\&Delta;}{s}_n=272.0106\\ \mathrm{\&Delta;}{s}_u=0\end{array}\right.\left(m\right),$ $\left\{\begin{array}{c}{v}_{e,m}=-83.9072\\ v_{n,m}=54.4021\\ v_{u,m}=0\end{array}\right.(m/s),$ $\left\{\begin{array}{c}{x}_m=-28.3400\\ y_m=54.4021\\ z_m=0\end{array}\right.\left(m\right)$

(b) for the 4th segmentation, carrier is from the last position (28.3400 of a upper motor segment,-95.8038,0) initial, more than the end speed rate of a motor segment be that first speed is accelerated circular motion clockwise, the surface level of plane of movement and aforementioned geographic coordinate system is 45 degree deflections (being γ=45), lasts 5s (being Δ t=5s), then at the initial time of this motor segment, the initial orientation of carrier, first speed, initial position are respectively:

φ ₀＝4.1372rad，v ₀＝100m/s， $\left\{\begin{array}{c}{x}_0^p=-28.3400\\ y_0^p=-95.8038\end{array}\right.\left(m\right)$

And in the finish time of this motor segment, the corner variable quantity of carrier, position angle, plane of movement speed and plane of movement coordinate are respectively:

Δφ＝-5rad，φ _m＝-0.8628rad， $\left\{\begin{array}{c}{v}_{x,m}=-227.9064\\ v_{y,m}=-195.0864\end{array}\right.(m/s),$ $\left\{\begin{array}{c}{x}_m^p=210.5218\\ y_m^p=-79.9270\end{array}\right.\left(m\right)$

Further realize the conversion to geographic coordinate system, then rotation of coordinate matrix, geographic coordinate system bearer rate and position coordinates are respectively:

$R_p^l=\left[\begin{array}{cc}1& 0\\ 0& 0.7071\\ 0& -0.7071\end{array}\right],$ $\left\{\begin{array}{c}{v}_{e,m}=-227.9064\\ v_{n,m}=-137.9469\\ v_{u,m}=137.9469\end{array}\right.(m/s),$ $\left\{\begin{array}{c}{x}_m=210.5218\\ y_m=-56.5169\\ z_m=56.5169\end{array}\right.\left(m\right)$

(c) for the 5th segmentation, carrier is from the last position (210.5218 of a upper motor segment,-56.5169,56.5169) initial, with (227.9064 ,-137.9469,137.9469) initial velocity acceleration rectilinear motion, last 5s (being Δ t=5s), then in the finish time of this motor segment, change of distance amount, speed and the position coordinates of carrier are respectively:

$\left\{\begin{array}{c}\mathrm{\&Delta;}{s}_e=-327.0319\\ \mathrm{\&Delta;}{s}_n=-264.7344\\ \mathrm{\&Delta;}{s}_u=189.7344\end{array}\right.\left(m\right),$ $\left\{\begin{array}{c}{v}_{e,m}=97.0936\\ v_{n,m}=32.0531\\ v_{u,m}=-62.0531\end{array}\right.(m/s),$ $\left\{\begin{array}{c}{x}_m=-116.5101\\ y_m=-321.2514\\ z_m=246.2514\end{array}\right.\left(m\right)$

Step 6: each emulation epoch at simulator the carrier present position in the geographic coordinate system is transformed into earth coordinates, is used for user interface and shows in real time output.Because with WGS-84 terrestrial coordinate system example, the major parameter of earth ellipsoid is defined as follows among the WGS-84: semi-major axis a=6378137m, ellipticity f=1/298.257223563, then in finish time of above-mentioned motion segmentation, earth coordinates carrier positions coordinate and geographic coordinate system bearer rate that user interface shows in real time see Table 1; And be used for the corresponding carrier positions coordinate of ECEF coordinate system and the speed that the further function of support simulator realizes, see Table 2.

Earth coordinates carrier positions and the geographic coordinate system bearer rate of each motor segment of table 1 finish time

ECEF coordinate system carrier positions and the speed of each motor segment of table 2 finish time

Claims (1)

1. the method for generating universal three-dimensional carrier motion trail in the GNSS simulator, it is characterized in that: the method concrete steps are as follows:

Step 1: generate in advance carrier movement track user profile, configuration file has defined carrier movement segmentation and trajectory parameters setting;

The carrier initial position is set to the coordinate in the earth coordinates

Be longitude, latitude and elevation; The carrier initial velocity be set in the geographic coordinate system east orientation, north orientation and day to initial velocity (v _{E, 0}, v _{N, 0}, v _{U, 0});

About the trajectory parameters setting of carrier movement segmentation, for the rectilinear motion type, the trajectory parameters that needs to arrange comprises: the rectilinear motion type identification, and the motor segment duration, the east orientation acceleration, the north orientation acceleration, the sky is to acceleration; For the circular motion type, the trajectory parameters that needs to arrange comprises: circular motion type identification, motor segment duration, direction of motion is namely moved clockwise or counterclockwise, radius of a circle, linear acceleration, the deflection of the relative geographic coordinate system of circular motion in-plane;

Step 2: when the carrier movement trace simulation began, being longitude, latitude and elevation take the initial position that arranges set up geographic coordinate system, initial position (x as initial point ₀, y ₀, z ₀) be (0,0,0);

To be the e axle point to east along local parallel tangent line to the x axle of geographic coordinate system, the y axle be the n axle along local warp tangent line energized north, the z axle is that the u axle is that earth reference ellipsoid surface normal points to zenith along local geographic vertical; For the carrier movement trace simulation of dynamic range in 180km, the geographic coordinate system of this initial setting up is considered as three axis directions and remains unchanged; Caused error is ignored, and then initial geographic coordinate system will be held in this distance range; Before the carrier movement trace simulation finishes, if carrier dynamically exceeds this distance range, then need to upgrade geographic coordinate system;

Step 3: obtain the trajectory parameters of carrier movement the first segmentation, if be the space line type of sports, in geographic coordinate system, according to initial velocity (v _{E, 0}, v _{N, 0}, v _{U, 0}), acceleration (a _e, a _n, a _u) setting, determine present position, the speed of carrier;

Motor segment is in the duration, according to t _K-1Position (the x of emulation moment epoch carrier _K-1, y _K-1, z _K-1), speed (v _{E, k-1}, v _{N, k-1}, v _{U, k-1}) calculate t _kMoment carrier positions (x _k, y _k, z _k), speed (v _{E, k}, v _{N, k}, v _{U, k}) method as follows:

(a) computing time interval of delta t

Δt＝t _k-t _k-1

(b) calculate t _K-1The time be carved into t _kChange of distance amount (Δ s constantly _e, Δ s _n, Δ s _u)

$\left\{\begin{array}{c}\mathrm{\&Delta;}{s}_e=v_{e,k-1}\&CenterDot;\mathrm{\&Delta;t}+\frac{1}{2}{a}_e\&CenterDot;{\mathrm{\&Delta;t}}^2\\ \mathrm{\&Delta;}{s}_n=v_{n,k-1}\&CenterDot;\mathrm{\&Delta;t}+\frac{1}{2}{a}_n\&CenterDot;{\mathrm{\&Delta;t}}^2\\ \mathrm{\&Delta;}{s}_u=v_{u,k-1}\&CenterDot;\mathrm{\&Delta;t}+\frac{1}{2}{a}_u\&CenterDot;{\mathrm{\&Delta;t}}^2\end{array}\right.$

(c) calculate t _kMoment bearer rate (v _{E, k}, v _{N, k}, v _{U, k})

$\left\{\begin{array}{c}{v}_{e,k}=v_{e,k-1}+a_e\&CenterDot;\mathrm{\&Delta;t}\\ v_{n,k}=v_{n,k-1}+a_n\&CenterDot;\mathrm{\&Delta;t}\\ v_{u,k}=v_{u,k-1}+a_u\&CenterDot;\mathrm{\&Delta;t}\end{array}\right.$

(d) calculate t _kMoment carrier positions (x _k, y _k, z _k)

$\left\{\begin{array}{c}{x}_k=x_{k-1}+{\mathrm{\&Delta;s}}_e\\ y_k=y_{k-1}+\mathrm{\&Delta;}{s}_n\\ z_k=z_{k-1}+\mathrm{\&Delta;}{s}_u\end{array}\right.$

Step 4: if carrier movement first is segmented into space circumference type of sports, namely get the angle of plane of movement direction and geographic coordinate system z axle according to the relative geographic coordinate system deflection of plane of movement γ, setting take z axle right side as positive-angle, at first determine the two dimensional surface of circular motion in the geographic coordinate system, the x axle on this plane is geographic coordinate system x axle, and the y axle is vertical with the x axle;

Secondly, according to initial velocity (v _{E, 0}, v _{N, 0}, v _{U, 0}), the setting of radius of a circle R and linear acceleration a, determine present position, the speed of carrier in the circular motion plane; Motor segment calculates t in the duration _kCarrier positions in the emulation moment epoch circular motion plane

Speed (v _{X, k}, v _{Y, k}) method as follows:

(a) computing time interval of delta t

Δt＝t _k-t _k-1

(b) calculate t _K-1The time be carved into t _kCorner variation delta φ constantly

$\mathrm{\&Delta;\&phi;}=-\mathrm{sign}\&CenterDot;(\frac{\left|v_{k-1}\right|}{R}\&CenterDot;\mathrm{\&Delta;t}+\frac{1}{2}\&CenterDot;\frac{a}{R}\&CenterDot;\mathrm{\&Delta;}{t}^2)$

Wherein, | .| is absolute value operators,

And $v_{k-1}=\sqrt{v_{e,k-1}^2+v_{n,k-1}^2+v_{u,k-1}^2}$

(c) calculate t _kCarrier position angle φ in the moment plane of movement _k

φ _k＝φ _k-1+Δφ

And initial orientation ${\mathrm{\&phi;}}_{k-1}=\left\{\begin{array}{cc}\arctan \frac{v_{x,k-1}}{v_{y,k-1}},& \frac{v_{x,k-1}}{v_{y,k-1}}>0\\ \mathrm{\&pi;}-\arctan \frac{v_{x,k-1}}{v_{y,k-1}},& \frac{v_{x,k-1}}{v_{y,k-1}}<0\end{array}\right.$

(d) calculate t _kBearer rate (v in the moment plane of movement _{X, k}, v _{Y, k})

$\left\{\begin{array}{c}{v}_{x,k}=\mathrm{sign}\&CenterDot;\left|v_k\right|\&CenterDot;\sin {\mathrm{\&phi;}}_k\\ v_{y,k}=-\mathrm{sign}\&CenterDot;\left|v_k\right|\&CenterDot;\cos {\mathrm{\&phi;}}_k\end{array}\right.$

And | v _k|=| v _K-1|+a Δ t

(e) calculate t _kCarrier positions in the moment plane of movement

$\left\{\begin{array}{c}{x}_k^p=\mathrm{sign}\&CenterDot;R\cos {\mathrm{\&phi;}}_k+\mathrm{sign}\&CenterDot;\frac{v_{y0}\&CenterDot;R}{\sqrt{v_{x0}^2+v_{y0}^2}}+x_0^p\\ y_k^p=\mathrm{sign}\&CenterDot;R\sin {\mathrm{\&phi;}}_k-\mathrm{sign}\&CenterDot;\frac{v_{x0}\&CenterDot;R}{\sqrt{v_{x0}^2+v_{y0}^2}}+y_0^p\end{array}\right.$

And $\left[\begin{array}{c}{x}_0^p\\ y_0^p\end{array}\right]=\left[\begin{array}{ccc}1& 0& 0\\ 0& \cos \mathrm{\&gamma;}& -\sin \mathrm{\&lambda;}\end{array}\right]\&CenterDot;\left[\begin{array}{c}{x}_0\\ y_0\\ z_0\end{array}\right]$

Wherein, λ is longitude corresponding to carrier coordinate; At last, with the position of carrier in the plane of movement and the rate conversion corresponding sports state parameter in the geographic coordinate system, conversion method is as follows:

(a) determine that plane of movement coordinate system p is to the rotation of coordinate matrix of geographic coordinate system L

$R_p^L=\left[\begin{array}{cc}1& 0\\ 0& \cos \mathrm{\&gamma;}\\ 0& -\sin \mathrm{\&gamma;}\end{array}\right]$

(b) calculate t _kMoment geographic coordinate system bearer rate (v _{E, k}, v _{N, k}, v _{U, k})

$\left[\begin{array}{c}{v}_{e,k}\\ v_{n,k}\\ v_{u,k}\end{array}\right]=R_p^L\&CenterDot;\left[\begin{array}{c}{v}_{x,k}\\ v_{y,k}\end{array}\right]$

(c) calculate t _kMoment geographic coordinate system carrier positions (x _k, y _k, z _k)

$\left[\begin{array}{c}{x}_k\\ y_k\\ z_k\end{array}\right]=R_p^L\&CenterDot;\left[\begin{array}{c}{x}_k^p\\ y_k^p\end{array}\right]$

Step 5: the user according to straight line or circular motion type arranges, and according to step 3, step 4, processes successively the subsequent motion segmentation; For the linking between the motion segmentation, more than the last current state of a motor segment be the initial equilibrium state of current motor segment, namely go up the carrier end position (x of a motor segment _m, y _m, z _m), end speed (v _{E, m}, v _{N, m}, v _{U, m}) become respectively the initial position (x of current motor segment ₀, y ₀, z ₀), initial velocity (v _{E, 0}, v _{N, 0}, v _{U, 0}), namely

$\left[\begin{array}{c}{x}_m\\ y_m\\ z_m\end{array}\right]=\left[\begin{array}{c}{x}_0\\ y_0\\ z_0\end{array}\right],$ $\left[\begin{array}{c}{v}_{e,m}\\ v_{n,m}\\ v_{u,m}\end{array}\right]=\left[\begin{array}{c}{v}_{e,0}\\ v_{n,0}\\ v_{u,0}\end{array}\right]$

Step 6: each emulation epoch at simulator the carrier present position in the geographic coordinate system is transformed into earth coordinates, is used for user interface and shows in real time output; And be that the ECEF coordinate system is judged with further support satellite visibility with carrier present position, rate conversion to earth rectangular coordinate system, pseudorange, Doppler signal state parameter compute simulator function realize;

(a) calculate t _kCorresponding carrier positions coordinate in the moment earth coordinates

Wherein,

A is earth reference ellipsoid semi-major axis, and e is earth reference ellipsoid excentricity;

Be convenient to for the purpose of the numerical evaluation, consider R _NAnd R _MApproximate data, namely

And ellipticity $f=1-\sqrt{1-e^2};$

(b) calculate t _kCorresponding carrier positions coordinate (x in the moment ECEF coordinate system _{E, k}, y _{E, k}, z _{E, k})

Wherein, λ,

Be carrier positions coordinate (x _k, y _k, z _k) corresponding warp, latitude, and (x _Oe, y _Oe, z _Oe) be the geographic coordinate system initial point

Corresponding coordinate in the ECEF coordinate system calculates by following formula:

And

(c) calculate t _kCorresponding bearer rate (v in the moment ECEF coordinate system _{X, k}, v _{Y, k}, v _{Z, k})

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