CN104359454A - Atmospheric polarized light-based solar space position acquisition method - Google Patents

Abstract

The invention relates to solar space position acquiring technology, and particularly relates to an atmospheric polarized light distribution mode-based solar space position acquisition method, which comprises the following steps: determining a sun as a space characteristic point through analyzing the atmospheric polarization mode, acquiring limited point sampling information of the atmospheric polarization mode by utilizing a polarized light detection sensor array on the basis of establishment of three-dimensional space coordinate system of the atmospheric polarization mode, establishing a solar space position information-containing equation set by combining Rayleigh scattering theory-based atmospheric polarization mode theoretical model, solving the equation to obtain the solar space position by combining Gaussian newton iteration expression through utilizing least square method. The sun serves as a reliable space characteristic point in the polarization mode, the acquisition of space positions has wide application prospects in the fields of polarized light navigation, solar tracking and the like.

Description

Based on the solar space location acquiring method of atmospheric polarization light

Technical field

The present invention relates to a kind of solar space location acquiring method, be specially a kind of method for solving of the solar space position based on atmospheric polarization light distribution pattern, be with a wide range of applications in the field such as Solar use, polarotactic navigation.

Background technology

Towards the technical need such as space characteristics point measurement in the solar tracking of field of solar energy utilization, polarotactic navigation field, based on the rayleigh model describing atmospheric polarization type, the present invention proposes a kind of solar space location acquiring method based on atmospheric polarization light, in the field such as Solar use and polarotactic navigation, there is important using value.

In field of solar energy utilization, because sun power has the feature such as dispersiveness, instability, therefore the study hotspot that photovoltaic efficiency becomes Chinese scholars is improved, one of them important channel adopts sun real time technique for tracking exactly, the method that the sun is followed the tracks of in current realization mainly comprises photoelectric tracking and looks daily motion track following two kinds, be opened loop control process depending on daily motion track following, there is cumulative errors and cannot revise it, require high to tracking means installation accuracy in addition; Photoelectric tracking adopts closed-loop control form, its tracking sensitivity is higher, and the site error that can not produce when initially installing by tracking means affects, but because light activated element easily affects by weather conditions and environment spurious rays, cause the malfunction of follower, be thus difficult to the Stability and veracity ensureing to follow the tracks of.So, the research direction be detected as in order to current solar tracking technology how to reduce weather effect, realizing to position of sun more accurately and stably.

In polarotactic navigation field, existing polarotactic navigation technique study, the two-dimentional course information all concentrating on carrier obtains, situation about all can only move in the horizontal plane for navigation carrier, and the carrier navigation information acquisition of reality is not limited to two-dimentional course information in most cases, and on antijamming capability, symmetry due to sky polarization mode is easier to be subject to cloud layer etc., and to block impact destroyed, therefore utilizes the symmetric navigate mode of atmospheric polarization type still to there is larger defect.Resolve application aspect at polarized light three-dimensional attitude and have correlative study, its core is the determination of space characteristics point position, in atmospheric polarization type, optional feature point comprises the sun, neutral point and zenith point, wherein neutral point and zenith point all cannot provide effective dynamic reference positional information, so utilize atmospheric polarization distribution pattern determination solar space position to be a key problem in polarotactic navigation theory, and there is no the correlative study utilizing atmospheric polarization type simultaneously to determine solar space position height, azimuth information method.

In sum, in field of solar energy utilization and polarotactic navigation field, all there is many restrictions with not enough in prior art, especially the environmental factor change such as cloud cover obtains impact very greatly accurately on solar space position stability, and this impact cannot effectively be eliminated.

Summary of the invention

The present invention is directed to the theoretical and field of solar energy utilization of polarotactic navigation to the demand of space characteristics space of points positional information, object is to provide a kind of method utilizing atmospheric polarization type acquisition solar space positional information newly.What the present invention utilized atmospheric polarization type is distributed in the characteristic in whole spatial domain with high stability, and in relative two fields, existing method has stronger antijamming capability and robustness, is with a wide range of applications.

The present invention adopts following technical scheme to realize:

Based on a solar space location acquiring method for atmospheric polarization light, comprise the steps:

A, set up and describe carrier and measure the three-dimensional coordinate system OXYZ of atmospheric polarization type, coordinate system is with carrier center of gravity for initial point O, OX axle points on the right side of carrier, and OY axle is along carrier body direction of principal axis and point to front, and OZ axle perpendicular to carrier straight up; Polarization information on all directions all describes on the hemisphere face taking O as initial point.

Above-mentioned hemisphere face to distribute n sampled point P altogether by Arbitrary distribution mode, and n>=2, the spherical co-ordinate position of each sampled point P is known, and on sphere, the directional information of i-th sampled point P is denoted as (h _i, α _i), h _ifor elevation angle, α _ifor position angle, position angle is the axial OY axle of 0, OX with OX axle is just.

In each sampled point P position, a polarized light detection sensor is installed, determines degree of polarization d and the polarization angle γ of this sampled point P at any time.

B, the polarization information of sampled points some in atmospheric polarization type to be sampled, sampling point position sequence is denoted as array P, i-th sampled point P _idimensional orientation coordinate be (h _i, α _i), the sampling of sample information composed atmosphere polarization mode array, i.e. degree of polarization or the polarization direction of each sampled point,

D＝(d ₁，d ₂，…，d _n)..............................................(4)

γ＝(γ ₁，γ ₂，…，γ _n)................................................(5)

Wherein i ∈ [1, n].

Sun altitude h is represented respectively with DOP and AOP _s, solar azimuth a _s, sampled point elevation angle h _i, sampled point azimuth angle alpha _ifour independents variable are with the funtcional relationship between degree of polarization d and polarization angle γ, and position of sun is denoted as S=(h _s, a _s), namely solved the system of equations of position of sun as follows,

$\left\{\begin{array}{c}{d}_1=\mathrm{DOP}(S,h_1,{\mathrm{\α}}_1)\\ d_2=\mathrm{DOP}(S,h_2,{\mathrm{\α}}_2)\\ ...\\ d_n=\mathrm{DOP}(S,h_n,{\mathrm{\α}}_n)\end{array}\right....\left(6\right)$

$\left\{\begin{array}{c}{\mathrm{\γ}}_1=\mathrm{AOP}(S,h_1,{\mathrm{\α}}_1)\\ {\mathrm{\γ}}_2=\mathrm{AOP}(S,h_2,{\mathrm{\α}}_2)\\ ...\\ {\mathrm{\γ}}_n=\mathrm{AOP}(S,h_n,{\mathrm{\α}}_n)\end{array}\right....\left(7\right)$

Solving of c, system of equations

System of equations (6) is rewritten into following form:

f _i(S)＝DOP(S，h _i，α _i)-d _i.......................................................(8)

F _i(S) represent the residual function of degree of polarization, above-mentioned Nonlinear System of Equations is converted into least square problem, namely asks the minimum value of following objective function

$\min f\left(S\right)=\min \frac{1}{2}{\mathrm{\Σ}}_{i=1}^n{f_i}^2\left(S\right)=\min \frac{1}{2}{\left|\right|F\left(S\right)\left|\right|}_2^2...\left(9\right)$

Wherein, F (S)=[f ₁(S) f ₂(S) ... f _n(S)] ^t, adopt Gauss-Newton (Gauss-Newton) method to solve, its iterative formula is

(J(S _k) ^TJ(S _k))ΔS _k＝-J(S _k)F(S _k)..............................(10)

S _k+1＝S _k+ΔS _k...................................................(11)

Wherein, J (S _k) be F (S _k) Jacobi (Jacobian) matrix, namely

$J\left(S_k\right)=\left[\begin{array}{cccc}\frac{{\∂f}_1\left(S_k\right)}{{\∂S}_k\left(1\right)}& \frac{{\∂f}_2\left(S_k\right)}{{\∂S}_k\left(1\right)}& ...& \frac{{\∂f}_n\left(S_k\right)}{{\∂S}_k\left(1\right)}\\ \frac{{\∂f}_1\left(S_k\right)}{{\∂S}_k\left(2\right)}& \frac{{\∂f}_2\left(S_k\right)}{{\∂S}_k\left(2\right)}& ...& \frac{{\∂f}_n\left(S_k\right)}{{\∂S}_k\left(2\right)}\end{array}\right]...\left(12\right)$

The detailed process of iteration is:

(1), make k=0, choose initial value S ₀, stop error ε > 0, and be a very little constant;

(2) Jacobian matrix J (S, is calculated _k), substitute into equation (10) and solve Δ S _k;

(3), g is made _k=J (S _k) F (S _k), if || gk|| < ε, then stop iteration, otherwise turn next step;

(4), S is solved according to formula (11) _k+1, and return second step continuation iteration;

Thus, namely resolve and obtain solar space location parameter S=(h _s, a _s).

In like manner, or pass through said method solving equation group (7), solar space location parameter S=(h can be obtained equally _s, a _s).

Atmospheric polarization type contains abundant navigation information, utilize the distribution characteristics of atmospheric polarization type can provide attitude needed for navigation and course information for carrier, and the spatial positional information how utilizing atmospheric polarization light distributed model to obtain space characteristics point is the key realized based on the three-dimensional attitude acquisition method of atmospheric polarization type.The present invention is the solar tracking of field of solar energy utilization, the space characteristics point measurement in polarotactic navigation field provides a kind of acquisition methods reliably, by analyse atmos polarization mode, the sun is defined as space characteristics point, on the basis establishing atmospheric polarization type three-dimensional coordinate system, polarized light detection sensor array is utilized to obtain the finite point sample information of atmospheric polarization type, the system of equations comprising solar space positional information is set up in conjunction with the atmospheric polarization type theoretical model based on Rayleigh scattering theory, least square method is utilized to obtain solar space position in conjunction with Gauss-Newton Methods solving equation, for the navigation application of atmospheric polarization light is laid a good foundation.

Contrast with existing correlation technique, the advantage that the present invention has is as follows:

1, existing polarized light three-dimensional attitude acquisition method utilizes zenith point or neutral point as the space characteristics point of atmospheric polarization type, and by analyzing these two points in known practical application all cannot provide effective reference position information for three-dimensional posture fixing above, the present invention is directed to above-mentioned deficiency, studying on the basis based on the atmospheric polarization distributed model of Rayleigh scattering, a kind of method utilizing limited sampling dot information acquisition solar space positional information in atmospheric polarization type is proposed, for the realization of the three-dimensional attitude acquisition method utilizing atmospheric polarization light is laid a good foundation.

2, the present invention utilizes the atmospheric polarization type sample information in whole day territory to resolve in conjunction with optimization method and obtains solar space position, effectively make use of the polarized light distribution pattern of all-sky, the interference of the shelters such as cloud layer can utilize cloud layer recognition method by its filtering, therefore has high antijamming capability.

The present invention is reasonable in design, by analyse atmos polarization mode, the sun is defined as space characteristics point, on the basis establishing atmospheric polarization type three-dimensional coordinate system, polarized light detection sensor array is utilized to obtain the finite point sample information of atmospheric polarization type, the system of equations comprising solar space positional information is set up in conjunction with the atmospheric polarization type theoretical model based on Rayleigh scattering theory, least square method is utilized to obtain solar space position in conjunction with Gaussian weighting marks mode solving equation, the sun is as space characteristics point reliable in polarization mode, the acquisition of its locus is at polarotactic navigation, the fields such as solar tracking have wide practical use.

Accompanying drawing explanation

Fig. 1 represents various point locations relation schematic diagram under carrier coordinate system.

Fig. 2 represents polarization information sampling point distributions schematic diagram.

Embodiment

Below specific embodiments of the invention are described in detail.

Utilize atmospheric polarization type to obtain a method for solar space position, specifically comprise the steps:

A, as shown in Figure 1, sets up and describes the three-dimensional coordinate system OXYZ that carrier measures atmospheric polarization type, and coordinate system is with carrier center of gravity for initial point O, OX axle points on the right side of carrier, and OY axle is along carrier body direction of principal axis and point to front, and OZ axle perpendicular to carrier straight up.

Polarization information on all directions all describes on the hemisphere face taking O as initial point; Above-mentioned hemisphere face can to distribute n sampled point P altogether by Arbitrary distribution mode, and n>=2, the spherical co-ordinate position of each sampled point P is known, and on sphere, the directional information of i-th sampled point P is denoted as (h _i, α _i), h _ifor elevation angle, α _ifor position angle, position angle is the axial OY axle of 0, OX with OX axle is just; Sampled point can not limit the concrete distribution mode adopted herein, but preferably, on hemisphere face, the distribution mode of n sampled point P is for being distributed on each roundlet, as shown in Figure 2, and n sampled point P ₁~ P _nbe distributed in hemispherical 1st circle from top to bottom on the n-th circle; Definition: go to cut above-mentioned hemisphere by a plane, cross section is disc; The line in the centre of sphere and the center of circle, cross section is perpendicular to cross section; Sphere is called great circle by the circle that the plane through the centre of sphere intercepts, and is all called roundlet by the circle that the some cross sections without the centre of sphere intercept.

In each sampled point P position, a bionical polarized light detection sensor can be installed, measure the degree of polarization d obtaining this sampled point P and polarization angle γ at any time; That is, the polarization information (degree of polarization d and polarization angle γ) of sampled point P can be characterized by Rayleigh polarization model of the atmosphere, and namely degree of polarization d is calculated by following formula,

$d\left(\mathrm{\&theta;}\right)=\frac{1-{\cos }^2\mathrm{\&theta;}}{1+c{\mathrm{os}}^2\mathrm{\&theta;}}{P}_{\max }...\left(1\right)$

Wherein θ is scattering angle, can be tried to achieve by following formula

cosθ＝sin(h)sin(h _s)+cos(h)cos(h _s)cos(α _s-α)........(2)

Polarization angle γ is solved by following formula

$\mathrm{\&gamma;}={\tan }^{-1}\left(\frac{\cos \left(h\right)\sin \left(h_s\right)-\sin \left(h\right)\cos \left(h_s\right)\cos ({\mathrm{\&alpha;}}_s-\mathrm{\&alpha;})}{\sin ({\mathrm{\&alpha;}}_s-\mathrm{\&alpha;})\cos h_s}\right)...\left(3\right)$

B, the polarization information of sampled points some in atmospheric polarization type to be sampled, all sampling point position sequences are denoted as array P, i-th sampled point P _idimensional orientation coordinate be (h _i, α _i).The sampling of sample information (degree of polarization d and polarization angle γ) composed atmosphere polarization mode array, i.e. degree of polarization or the polarization direction of each sampled point,

D＝(d ₁，d ₂，…，d _n).................................................(4)

γ＝(γ ₁，γ ₂，…，γ _n)...................................................(5)

Wherein i ∈ [1, n].

The attitude information of sample information array and correspondence is substituted into formula (1), (2), (3), represents sun altitude h respectively with DOP and AOP _s, solar azimuth a _s, be sampled an elevation angle h _i, be sampled an azimuth angle alpha _ifour independents variable are with the funtcional relationship between degree of polarization and polarization angle, and position of sun is denoted as S=(h _s, a _s).The system of equations of position of sun can be solved as follows,

$\left\{\begin{array}{c}{d}_1=\mathrm{DOP}(S,h_1,{\mathrm{\&alpha;}}_1)\\ d_2=\mathrm{DOP}(S,h_2,{\mathrm{\&alpha;}}_2)\\ ...\\ d_n=\mathrm{DOP}(S,h_n,{\mathrm{\&alpha;}}_n)\end{array}\right....\left(6\right)$

$\left\{\begin{array}{c}{\mathrm{\&gamma;}}_1=\mathrm{AOP}(S,h_1,{\mathrm{\&alpha;}}_1)\\ {\mathrm{\&gamma;}}_2=\mathrm{AOP}(S,h_2,{\mathrm{\&alpha;}}_2)\\ ...\\ {\mathrm{\&gamma;}}_n=\mathrm{AOP}(S,h_n,{\mathrm{\&alpha;}}_n)\end{array}\right....\left(7\right)$

Because system of equations (6) or (7) comprise a series of sin cos functions, direct solution is very difficult, and usual equation quantity n > 2, make (6), (7) be over-determined systems, so cannot directly try to achieve its algebraic solution, numerical method need be adopted to solve.For system of equations (6), solution procedure is described.

Solving of c, system of equations

System of equations (6) is rewritten into following form:

f _i(S)＝DOP(S，h _i，α _i)-d _i..........................................................(8)

F _i(S) represent the residual function of degree of polarization, above-mentioned Nonlinear System of Equations is converted into least square problem, namely asks the minimum value of following objective function

$\min f\left(S\right)=\min \frac{1}{2}{\mathrm{\&Sigma;}}_{i=1}^n{f_i}^2\left(S\right)=\min \frac{1}{2}{\left|\right|F\left(S\right)\left|\right|}_2^2...\left(9\right)$

Wherein, F (S)=[f ₁(S) f ₂(S) ... f _n(S)] ^t, adopt Gauss-Newton (Gauss-Newton) method to solve, its iterative formula is

(J(S _k) ^TJ(S _k))ΔS _k＝-J(S _k)F(S _k)................................(10)

S _k+1＝S _k+ΔS _k......................................................(11)

Wherein, J (S _k) be F (S _k) Jacobi (Jacobian) matrix, namely

$J\left(S_k\right)=\left[\begin{array}{cccc}\frac{{\&PartialD;f}_1\left(S_k\right)}{{\&PartialD;S}_k\left(1\right)}& \frac{{\&PartialD;f}_2\left(S_k\right)}{{\&PartialD;S}_k\left(1\right)}& ...& \frac{{\&PartialD;f}_n\left(S_k\right)}{{\&PartialD;S}_k\left(1\right)}\\ \frac{{\&PartialD;f}_1\left(S_k\right)}{{\&PartialD;S}_k\left(2\right)}& \frac{{\&PartialD;f}_2\left(S_k\right)}{{\&PartialD;S}_k\left(2\right)}& ...& \frac{{\&PartialD;f}_n\left(S_k\right)}{{\&PartialD;S}_k\left(2\right)}\end{array}\right]...\left(12\right)$

The detailed process of iteration is:

(1), make k=0, choose initial value S ₀, stop error ε > 0, and be a very little constant;

(2) Jacobian matrix J (S, is calculated _k), substitute into equation (10) and solve Δ S _k;

(3), g is made _k=J (S _k) F (S _k), if || g _k|| < ε, then stop iteration, otherwise turn next step;

(4), S is solved according to formula (11) _k+1, and return second step continuation iteration.

Thus, namely resolve and obtain solar space location parameter S=(h _s, a _s).

In like manner, or by above-mentioned same method solving equation group (7), solar space location parameter S=(h can also be obtained _s, a _s).Certainly, the solar space location parameter obtained by system of equations (6) can't be identical with the solar space location parameter obtained by system of equations (7), but within rational actual error scope.

Claims (3)

1., based on a solar space location acquiring method for atmospheric polarization light, it is characterized in that: comprise the steps:

A, set up and describe carrier and measure the three-dimensional coordinate system OXYZ of atmospheric polarization type, coordinate system is with carrier center of gravity for initial point O, OX axle points on the right side of carrier, and OY axle is along carrier body direction of principal axis and point to front, and OZ axle perpendicular to carrier straight up; Polarization information on all directions all describes on the hemisphere face taking O as initial point;

Above-mentioned hemisphere face to distribute n sampled point P altogether by Arbitrary distribution mode, and n>=2, the spherical co-ordinate position of each sampled point P is known, and on sphere, the directional information of i-th sampled point P is denoted as (h _i, α _i), h _ifor elevation angle, α _ifor position angle, position angle is the axial OY axle of 0, OX with OX axle is just;

In each sampled point P position, a polarized light detection sensor is installed, determines degree of polarization d and the polarization angle γ of this sampled point P at any time;

B, the polarization information of sampled points some in atmospheric polarization type to be sampled, sampling point position sequence is denoted as array P, i-th sampled point P _idimensional orientation coordinate be (h _i, α _i), the sampling of sample information composed atmosphere polarization mode array, i.e. degree of polarization or the polarization direction of each sampled point,

D＝(d ₁，d ₂，…，d _n).............................................(4)

γ＝(γ ₁，γ ₂，…，γ _n)................................................(5)

Wherein i ∈ [1, n].

Sun altitude h is represented respectively with DOP and AOP _s, solar azimuth a _s, be sampled an elevation angle h _i, be sampled an azimuth angle alpha _ifour independents variable are with the funtcional relationship between degree of polarization d and polarization angle γ, and position of sun is denoted as S=(h _s, a _s), namely solved the system of equations of position of sun as follows,

$\left\{\begin{array}{c}{d}_1=\mathrm{DOP}(S,h_1,{\mathrm{\&alpha;}}_1)\\ d_2=\mathrm{DOP}(S,h_2,{\mathrm{\&alpha;}}_2)\\ \&CenterDot;\&CenterDot;\&CenterDot;\\ d_n=\mathrm{DOP}(S,h_n,{\mathrm{\&alpha;}}_n)\end{array}\right....\left(6\right)$

$\left\{\begin{array}{c}{\mathrm{\&gamma;}}_1=\mathrm{AOP}(S,h_1,{\mathrm{\&alpha;}}_1)\\ {\mathrm{\&gamma;}}_2=\mathrm{AOP}(S,h_2,{\mathrm{\&alpha;}}_2)\\ \&CenterDot;\&CenterDot;\&CenterDot;\\ {\mathrm{\&gamma;}}_n=\mathrm{AOP}(S,h_n,{\mathrm{\&alpha;}}_n)\end{array}\right....\left(7\right)$

Solving of c, system of equations

System of equations (6) is rewritten into following form:

f _i(S)＝DOP(S，h _i，α _i)-d _i.....................................................(8)

F _i(S) represent the residual function of degree of polarization, above-mentioned Nonlinear System of Equations is converted into least square problem, namely asks the minimum value of following objective function

$\min f\left(S\right)=\min \frac{1}{2}{\mathrm{\&Sigma;}}_{i=1}^n{f}_i^2\left(S\right)=\min \frac{1}{2}{\left|\right|F\left(S\right)\left|\right|}_2^2...\left(9\right)$

Wherein, F (S)=[f ₁(S) f ₂(S) ... f _n(S)] ^t, adopt gauss-newton method to solve, its iterative formula is

(J(S _k) ^TJ(S _k))ΔS _k＝-J(S _k)F(S _k)..............................(10)

S _k+1＝S _k+ΔS _k...................................................(11)

Wherein, J (S _k) be F (S _k) Jacobian matrix, namely

$J\left(S_k\right)=\left[\begin{array}{cccc}\frac{{\&PartialD;f}_1\left(S_k\right)}{{\&PartialD;S}_k\left(1\right)}& \frac{{\&PartialD;f}_2\left(S_k\right)}{{\&PartialD;S}_k\left(1\right)}& \&CenterDot;\&CenterDot;\&CenterDot;& \frac{{\&PartialD;f}_n\left(S_k\right)}{{\&PartialD;S}_k\left(1\right)}\\ \frac{{\&PartialD;f}_1\left(S_k\right)}{{\&PartialD;S}_k\left(2\right)}& \frac{{\&PartialD;f}_2\left(S_k\right)}{{\&PartialD;S}_k\left(2\right)}& \&CenterDot;\&CenterDot;\&CenterDot;& \frac{{\&PartialD;f}_n\left(S_k\right)}{{\&PartialD;S}_k\left(2\right)}\end{array}\right]...\left(12\right)$

The detailed process of iteration is:

(1), make k=0, choose initial value S ₀, stop error ε > 0, and be a very little constant;

(2) Jacobian matrix J (S, is calculated _k), substitute into equation (10) and solve Δ S _k;

(3), g is made _k=J (S _k) F (S _k), if || g _k|| < ε, then stop iteration, otherwise turn next step;

(4), S is solved according to formula (11) _k+1, and return second step continuation iteration;

Thus, namely resolve and obtain solar space location parameter S=(h _s, a _s);

In like manner, or by said method solving equation group (7), solar space location parameter S=(h is obtained _s, a _s).

2. the solar space location acquiring method based on atmospheric polarization light according to claim 1, is characterized in that: in described step a, and on hemisphere face, the distribution mode of n sampled point P is for being distributed on each roundlet.

3. the solar space location acquiring method based on atmospheric polarization light according to claim 1 and 2, is characterized in that: in described step a, and the polarization information of each sampled point P can be characterized by Rayleigh polarization model of the atmosphere, and namely degree of polarization d is calculated by following formula,

$d\left(\mathrm{\&theta;}\right)=\frac{1-{\cos }^2\left(\mathrm{\&theta;}\right)}{1+{\cos }^2\mathrm{\&theta;}}{P}_{\max }...\left(1\right)$

Wherein θ is scattering angle, can be tried to achieve by following formula

cosθ＝sin(h)sin(h _s)+cos(h)cos(h _s)cos(α _s-α).........(2)

Polarization angle γ is solved by following formula

$\mathrm{\&gamma;}={\tan }^{-1}\left(\frac{\cos \left(h\right)\sin \left(h_s\right)-\sin \left(h\right)\cos \left(h_s\right)\cos ({\mathrm{\&alpha;}}_s-\mathrm{\&alpha;})}{\sin ({\mathrm{\&alpha;}}_s-\mathrm{\&alpha;})\cos h_s}\right)...\left(3\right).$