Forward dual-beam Doppler detection of aircraft roll angle

Abstract

The invention provides a method for measuring and calculating a roll angle of an aircraft by using a forward dual-beam Doppler navigation radar. The method is mainly characterized in that two coordinate systems are respectively used for analytical calculation. Firstly, a drift angle and an axial speed are solved in an airframe coordinate system. Then, in horizontal plane coordinate systems, a Doppler frequency shift equation when the aircraft is only in a rolling state is considered, and at the moment, the plane included angle between the horizontal planes of the two coordinate systems is the rolling angle of the aircraft. The analytic solution of the roll angle can be obtained by utilizing the relationship between the beam deflection angles of the two coordinate systems in the inclined vertical plane where the beam is located and performing reasonable approximate processing. At present, only a roll angle calculation method when no pitch angle exists in the side view surface of the aircraft is given. When the pitch angle exists in the side view surface of the aircraft, due to the fact that many unknown variables are involved, analysis and calculation of the roll angle of the aircraft need to be further analyzed and studied. However, the existing analysis result lays a reliable foundation for subsequent in-depth research.

Description

Technical Field

[0001] This invention relates to radio navigation technology, and more specifically to a method for calculating the roll angle of an aircraft using a forward dual-beam Doppler navigation radar. Background Technology

[0002] The aircraft's pitch and roll angles are provided by its attitude sensors. In 2023, an invention patent for a dual-beam Doppler navigation radar was filed [unpublished patent application, Yu Tao, "A Boundary Calculation Method for Doppler Navigation Parameters Using Only a Single-Directional Dual Beam," 202310314850.8, 2023-03-28], which significantly improved the design methods of existing Doppler navigation radars. Firstly, its academic contribution lies in correcting errors in the original technical methods, such as inconsistent measurement benchmarks and the independence of velocity components from the aircraft's pitch angle. Secondly, its breakthrough in research methodology involves performing secondary analytical calculations in both the body coordinate system and the horizontal coordinate system. Therefore, using only two beams, it is possible not only to obtain the three-dimensional velocity components but also to deduce the pitch angle, which was previously impossible to obtain using only these two beams.

[0003] Based on this, the present invention patent further provides a method for solving the roll angle of an aircraft based solely on Doppler frequency shift measurements. Summary of the Invention

[0004] This invention presents a method for calculating the roll angle of an aircraft using a forward-facing dual-beam Doppler navigation radar. Its main feature is the use of two separate coordinate systems for analytical calculation. First, the yaw angle and axial velocity are solved in the body coordinate system. Then, in the horizontal coordinate system, the Doppler frequency shift equation is considered when the aircraft is only in a roll state. The angle between the horizontal planes of the two coordinate systems is the roll angle. By utilizing the relationship between the beam deflection angles of the two coordinate systems within the oblique vertical plane containing the beam, and through reasonable approximations, the analytical solution for the roll angle can be obtained.

[0005] This invention is achieved through the following technical solution:

[0006] Doppler frequency shift data was detected using an airborne dual-beam Doppler navigation radar.

[0007] First, solve for the aircraft's yaw angle and axial velocity in the body coordinate system.

[0008] Furthermore, in the horizontal coordinate system, by utilizing the relationship between the beam deflection angles of the two coordinate systems within the oblique vertical plane where the beam is located, and by making reasonable approximations to the Doppler frequency shift equation in the horizontal coordinate system, the analytical solution for the roll angle can be obtained.

[0009] Specifically, the following steps are included:

[0010] Step 1: The aircraft has been equipped with a Doppler navigation radar. At this point, after the antenna feed equipment for the airborne Doppler radar has been installed, the beam deflection angle β in the horizontal plane of the aircraft coordinate system is... h and the beam deflection angle β in the vertical plane of the body coordinate system v All of these are known, fixed, and unchanging data.

[0011] Step 2: In the body coordinate system, the Doppler frequency shift obtained by each beam is:

[0012]

[0013]

[0014] In the formula: f d1 and f d2 It is the Doppler frequency shift; λ is the wavelength; v w δ is the longitudinal velocity of the aircraft; δ is the yaw angle in the horizontal plane of the aircraft coordinate system; β h It is the beam deflection angle in the horizontal plane of the body coordinate system; β v It is the beam deflection angle in the vertical plane of the body coordinate system.

[0015] First, eliminate the axial velocity v by the ratio of these two Doppler frequency shifts. w Solve for the heading deviation angle:

[0016]

[0017] Then, subtract the two Doppler frequency shift formulas to solve for the longitudinal velocity of the aircraft:

[0018]

[0019] Step 3: In the horizontal coordinate system, without any pitch angle, the Doppler frequency shift equations obtained from the two beams in a rigorous mathematical form are as follows:

[0020]

[0021]

[0022] In the formula: v p It is the resultant velocity on the horizontal plane of the horizontal coordinate system; δ p It is the heading angle on the horizontal plane in the horizontal coordinate system; β hp It is the deflection angle of the beam in the horizontal plane; β vp It is the deflection angle of the beam in the vertical plane.

[0023] Step 4: Based on the geometric relationships, give the relationships between the various parameters:

[0024] (1) The following relationship exists between the deflection angle and the resultant velocity in the two horizontal planes of the two coordinate systems:

[0025] v w cosδ=v p cosδ p (7)

[0026] Because of δ p When ≈δ, therefore we have:

[0027] v p ≈v w (8)

[0028] (2) The relationship between the beam deflection angles in the two coordinate systems within the oblique vertical plane where the beam is located:

[0029]

[0030] In the formula: α rβ It is the roll angle within the oblique vertical plane where the beam is located.

[0031] (3) The relationship between the beam deflection angles in the horizontal plane of the two coordinate systems:

[0032]

[0033] Step 5: Based on the relationships given in Step 4, rewrite the frequency shift equation in the horizontal coordinate system as follows:

[0034]

[0035]

[0036] Step 6: Expand and subtract the two frequency shift equations (11) and (12):

[0037]

[0038]

[0039] Step 7: After rearranging parts of equations (13) and (14), eliminate sinβ using the ratio of the two equations. hp :

[0040]

[0041] Step 8: Expanding equation (15) yields a two-variable linear equation:

[0042]

[0043] From this, the roll angle in the oblique vertical plane can be calculated:

[0044]

[0045] in:

[0046] a=cosβ h cosδsinβ v ·tgβ v

[0047]

[0048]

[0049] Step 9: Further utilize the angle calculation formula for a regular triangular pyramid to obtain the roll angle in the vertical plane:

[0050]

[0051] In the formula: α r It is the roll angle in the vertical plane. Attached Figure Description

[0052] Figure 1 Geometric model

[0053] Figure 2 Relative calculation error of roll angle Detailed Implementation

[0054] The following is in conjunction with the appendix Figure 1 and attached Figure 2 The invention will be further explained in terms of how it is implemented.

[0055] Example

[0056] A method for detecting the roll angle of an aircraft using an airborne Doppler navigation radar. (Attached) Figure 1 It is a geometric model; attached Figure 2 It is the relative calculation error of the roll angle.

[0057] The technical analysis provided in the filed invention patent [unpublished patent application, Yu Tao, "A Boundary Calculation Method for Doppler Navigation Parameters Using Only a Single-Directional Dual-Beam Array", 202310314850.8, 2023-03-28] shows that the pitch angle based on frequency shift detection can be obtained simply by using a dual-coordinate system for analytical derivation. The main process is as follows: Firstly, the pitch angle is introduced by using two coordinate systems, because the angle between the horizontal plane of the body coordinate system and the horizontal plane coordinate system is the aircraft's pitch angle. Secondly, the influence of the aircraft's pitch angle is considered in the Doppler frequency shift equation. Furthermore, although the unknown aircraft pitch angle is added to the Doppler frequency shift equation, four boundary equations can be obtained through secondary analytical calculations in the body coordinate system and the horizontal plane coordinate system. The geometric relationship between the two coordinate systems can also be utilized. This creates conditions for solving problems with more than two unknown variables using only two beams.

[0058] This invention again utilizes a dual-coordinate system for analysis and calculation, presenting a method for directly determining the aircraft's roll angle using Doppler frequency shift detection. Currently, only a method for calculating the roll angle when there is no pitch angle within the aircraft's side view is presented. When there is a pitch angle within the aircraft's side view, due to the involvement of numerous unknown variables, the analysis and calculation of the aircraft's roll angle require further in-depth research. However, the existing analytical results lay a reliable foundation for subsequent in-depth research.

[0059] I. Basic Geometric Framework

[0060] Assuming the aircraft has no pitch forward or backward during flight, but experiences roll left or right, as shown in the attached diagram. Figure 1 As shown in the figure. The double-dotted line represents the horizontal plane in the horizontal coordinate system, and the dotted line represents the horizontal plane in the body coordinate system.

[0061] Assume two beams are angled forward, with the left angled beam represented by a thick line segment BG1. For clarity, the right angled beam is not shown. When the aircraft rolls left or right, the ground projection point of the beams will shift; the ground projection point G1 of the left angled beam in the diagram is only a schematic sketch.

[0062] 1. The relationship between the resultant velocities in the two planes

[0063] The resultant velocity v on the horizontal plane within the horizontal coordinate system p There is a deflection angle δ p The resultant velocity v on the horizontal plane of the body coordinate system. w There is a deflection angle δ, which is not marked for clarity.

[0064] In the horizontal coordinate system, we have:

[0065] vy =v p cosδ p (1)

[0066] In the formula: v y The velocity component along the y-axis in the horizontal coordinate system; v p It is the resultant velocity on the horizontal plane in the horizontal coordinate system; δ p It is the heading deflection angle on the horizontal plane of the horizontal coordinate system.

[0067] In the body coordinate system, we have:

[0068] v yw =v w cosδ (2)

[0069] In the formula: v yw It is the velocity component along the y-axis in the body coordinate system.

[0070] When the aircraft is flying horizontally, the following should be true: v y =v yw Therefore, there is a relationship between the two coordinate systems:

[0071] v w cosδ=v p cosδ p (3)

[0072] If we make an approximation: δ p If ≈δ, then we can obtain:

[0073] v p ≈v w (4) 2. Roll angle is set as follows: ∠EBH = α r It is the roll angle in the positive vertical plane; BH = a; EH = b, then:

[0074]

[0075] Further assume: ∠FBI=α rβ It is the roll angle within the oblique vertical plane where the beam is located; BI = a * FI = b. Based on geometric relationships, we have:

[0076] a = a * cos(90°-β h (5)

[0077] From this, we can obtain the relationship between the roll angle in the positive vertical plane and the roll angle in the oblique vertical plane:

[0078]

[0079] 3. Deflection angle

[0080] As attached Figure 1 As shown, β hp It is the deflection angle of the beam on the horizontal plane within the horizontal coordinate system; β vp β is the beam deflection angle in the vertical plane of the horizontal coordinate system. h It is the beam deflection angle on the horizontal plane within the body coordinate system; β v It is the deflection angle of the beam in the vertical plane of the body coordinate system.

[0081] Using the angle calculation formula of a regular triangular pyramid, the deflection angle β on the horizontal plane within the body coordinate system can be determined. h and the roll angle α in the oblique vertical plane rβ Find the deflection angle β in the horizontal plane of the horizontal coordinate system. hp :

[0082]

[0083] II. Boundary Analysis in the Body Coordinate System

[0084] 1. Frequency shift equation

[0085] In the body coordinate system, the Doppler frequency shift obtained by each beam is:

[0086]

[0087]

[0088] In the formula: f d1 and f d2 It is the Doppler frequency shift; λ is the wavelength; v w It is the axial velocity of the aircraft; β h It is the deflection angle of the beam in the horizontal plane; β v It is the deflection angle in the vertical plane; δ is the heading deflection angle.

[0089] Among them, beam deflection angle β h and β v These are all fixed values ​​known after the antenna feeder equipment was installed.

[0090] 2. Deflection angle

[0091] First, eliminate the axial velocity v using the ratio of the two Doppler frequency shifts. w :

[0092]

[0093] Therefore, the heading yaw angle in the body coordinate system can be calculated:

[0094]

[0095] 3. Vertical axis velocity

[0096] After obtaining the deflection angle, subtracting the two Doppler frequency shift formulas yields:

[0097]

[0098] The vertical velocity can be obtained from this:

[0099]

[0100] III. Boundary Analysis in the Horizontal Coordinate System

[0101] 1. Doppler frequency shift equation

[0102] In a horizontal coordinate system, assuming no pitch angle exists, the vertical velocity v should be... z =0. In strict mathematical form, we should have:

[0103]

[0104]

[0105] Where: β hp It is the deflection angle of the beam in the horizontal plane; β vp It is the deflection angle of the beam in the vertical plane.

[0106] Based on the preceding analysis, when δ p When ≈δ, we have: v p ≈v w Furthermore, based on geometric analysis, within the oblique vertical plane where the beam resides, there will be a relationship between the deflection angles in the vertical planes of the two coordinate systems:

[0107]

[0108] Therefore, the frequency shift equation can be rewritten as:

[0109]

[0110]

[0111] in:

[0112]

[0113] At this point, the roll angle in the inclined vertical plane can be solved directly using Matlab.

[0114] 2. Analytical Solution

[0115] First, expand the right-hand side of the two frequency shift formulas:

[0116]

[0117]

[0118] Subtract the two equations:

[0119]

[0120]

[0121] Repositioning:

[0122]

[0123]

[0124] Eliminate sinβ by the ratio of the two equations. hp :

[0125]

[0126] Expanding this gives us a two-variable linear equation:

[0127]

[0128] The final result is:

[0129]

[0130] in:

[0131] a=cosβ h cosδsinβ v ·tgβ v

[0132]

[0133]

[0134] 3. Roll angle in the vertical plane

[0135] Using formula (6), the roll angle in the vertical plane can be obtained:

[0136]

[0137] IV. Simulation

[0138] 1. Theoretical value of the frequency shift equation

[0139] This invention only simulates whether the roll angle is correct. The frequency shift equation, approximately derived in a horizontal coordinate system, is used as the theoretical value for the simulation calculation:

[0140]

[0141]

[0142] To simulate and verify the drift angle and axial velocity, the theoretical value of the Doppler frequency shift must be obtained using the Doppler shift equation derived in the body coordinate system. Simulation calculations confirmed that the formulas for calculating the drift angle and axial velocity are correct.

[0143] 2. Parameter settings

[0144] The selected parameters are only for simulation calculations and are irrelevant to actual engineering design.

[0145] (1) Based on existing engineering design data, select the beam deflection angle in the body coordinate system:

[0146] β = 80°

[0147] β v =75°

[0148] In the formula: β is the deflection angle between the beam and the longitudinal axis of the aircraft: β v It is the deflection angle in the vertical plane.

[0149] Using the angle calculation formula for a regular triangular pyramid, the deflection angle in the horizontal plane of the body coordinate system can be obtained:

[0150]

[0151] (2) The heading deviation angle is artificially defined to vary continuously within a small range [-5°, 5°].

[0152] (3) The resultant velocity on the horizontal plane in the body coordinate system is artificially selected: v w =100m / s.

[0153] (4) The roll angle is preset artificially.

[0154] (5) The operating frequency can be arbitrarily selected, and the wavelength is determined by the relationship between speed, wavelength and frequency.

[0155] 3. Relative calculation error

[0156] First, calculate the theoretical value of the Doppler frequency shift using formulas (16) and (17). Then, directly calculate the roll angle in the oblique vertical plane using formula (26), and finally obtain the roll angle in the positive vertical plane using formula (27).

[0157] The relative formula for calculating the roll angle is:

[0158]

[0159] In the formula: α r0 This indicates the preset roll angle; α rj It is the roll angle obtained by simulating the velocity component.

[0160] Appendix Figure 2 The relative calculation error curves at different roll angles are given, and the calculation results show that the formula derived in this invention is correct.

Claims

1. A method for calculating the roll angle of an aircraft using a forward-facing dual-beam Doppler navigation radar, characterized by analytical calculations using two coordinate systems. First, two fundamental navigation parameters, yaw angle and axial velocity, are solved in the body coordinate system. Then, in the horizontal coordinate system, the Doppler frequency shift equation is considered when the aircraft is only in a roll state. The angle between the horizontal planes of the two coordinate systems is the roll angle of the aircraft. Once a reasonable approximation is made, the analytical solution for the roll angle can be obtained. The method includes the following steps: Step 1: The aircraft has been equipped with a Doppler navigation radar. At this point, after the antenna feed equipment for the airborne Doppler radar has been installed, the beam deflection angle β in the horizontal plane of the aircraft coordinate system is... h and the beam deflection angle β in the vertical plane of the body coordinate system v All of these are known, fixed, and unchanging data. Step 2: In the body coordinate system, the Doppler frequency shift obtained by each beam is: In the formula: f d1 and f d2 It is the Doppler frequency shift; λ is the wavelength; v w δ is the longitudinal velocity of the aircraft; δ is the yaw angle in the horizontal plane of the aircraft coordinate system; β h It is the beam deflection angle in the horizontal plane of the body coordinate system; β v It is the beam deflection angle in the vertical plane of the body coordinate system. First, eliminate the axial velocity v by the ratio of these two Doppler frequency shifts. w Solve for the heading deviation angle: Then, subtract the two Doppler frequency shift formulas to solve for the longitudinal velocity of the aircraft: Step 3: In the horizontal coordinate system, without any pitch angle, the Doppler frequency shift equations obtained from the two beams in a rigorous mathematical form are as follows: In the formula: v p It is the resultant velocity on the horizontal plane of the horizontal coordinate system; δ p It is the heading angle on the horizontal plane in the horizontal coordinate system; β hp It is the deflection angle of the beam in the horizontal plane; β vp It is the deflection angle of the beam in the vertical plane. Step 4: Based on the geometric relationships, give the relationships between the various parameters: (1) The following relationship exists between the deflection angle and the resultant velocity in the two horizontal planes of the two coordinate systems: in w cosδ=v p cosδ p (7) Because of δ p When ≈δ, therefore we have: in p ≈in w (8) (2) The relationship between the beam deflection angles in the two coordinate systems within the oblique vertical plane where the beam is located: In the formula: α rβ It is the roll angle within the oblique vertical plane where the beam is located. (3) The relationship between the beam deflection angles in the horizontal plane of the two coordinate systems: Step 5: Based on the relationships given in Step 4, rewrite the frequency shift equation in the horizontal coordinate system as follows: Step 6: Expand and subtract the two frequency shift equations (11) and (12): Step 7: After rearranging parts of equations (13) and (14), eliminate sinβ using the ratio of the two equations. hp : Step 8: Expanding equation (15) yields a two-variable linear equation: From this, the roll angle in the oblique vertical plane can be calculated: in: a=cosβ h cosδsinβ v ·tgβ v Step 9: Further utilize the angle calculation formula for a regular triangular pyramid to obtain the roll angle in the vertical plane: In the formula: α r It is the roll angle in the vertical plane.

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