A portable laser radar two-dimensional scanning system and scanning method based on single-axis rotating double circular wedge prism

Abstract

The present invention discloses a portable laser radar two-dimensional scanning system and scanning method based on a single-axis rotating double circular wedge prism, comprising two circular wedge prisms with a refractive index of n and a wedge angle of α, and two stepping motors. The values ​​of the refractive index n and the wedge angle α satisfy the critical total reflection condition at the last exit surface when the light beam is incident on the double circular wedge prism along the axis direction. By controlling the stepping motor by a computer to change the rotation angle of the double circular prism, when the light beam is incident on the double circular wedge prism along the axis direction, the exit light beam can be arbitrarily pointed within the range of 0 to 360° azimuth angle and 0 to 90°‑α zenith angle. The present invention enables the exit light beam to perform various mode space scanning such as PPI, RHI, DBS, etc. within a large range, which is used to detect the spatial distribution of atmospheric physical quantities.

Description

Technical Field

[0001] The invention belongs to the technical field of laser radar detection, and is applicable to a portable laser radar system which needs to perform light beam scanning measurement in order to obtain the spatial distribution of atmospheric physical quantities. Background Art

[0002] The two-dimensional scanning system is widely used in the fields of optoelectronic tracking, laser radar, especially wind laser radar, and is a key component of equipment such as wind laser radar and 3D scanning laser radar. At present, the mainstream system that can realize large-scale (hemispherical space) two-dimensional scanning of the outgoing beam is the XY two-axis two-dimensional scanning system. The system adopts a structure in which two 45° reflectors rotate around the horizontal axis and the vertical axis respectively. Although it can realize large-scale scanning, its structure is relatively complex and its volume is large, which is not conducive to the small-scale integrated integration of the entire optoelectronic system. The double prism rotating around a single axis can realize the arbitrary pointing of the outgoing beam within a certain range, and has the advantages of compact structure, fast adjustment speed, and small influence of mechanical transmission error on pointing accuracy. It can meet the actual application requirements of portable wind laser radar, 3D scanning aerosol laser radar for detecting the spatial distribution of aerosols, and other systems. Therefore, the research on controlling the pointing direction of the outgoing beam by rotating the wedge prism around a single axis has always attracted extensive attention from scholars at home and abroad.

[0003] From the literature survey, the existing research on the forward and reverse problems of single-axis rotating dual prisms mainly adopts methods such as first-order paraxial approximation, non-paraxial ray tracing and vector optical synthesis. The forward and reverse accurate analytical solutions obtained are very complex, which is not conducive to the optimization design and performance analysis of the actual system, and the approximate solution under small angles is not suitable for large-angle scanning. In addition, when analyzing the scanning method, there is no combination of scanning laser radar, especially wind laser radar used for detection of vector wind field, wind shear, aircraft wake, etc., and 3D scanning aerosol laser radar used for detection of aerosol cone and section distribution. Common scanning modes such as PPI, RHI, DBS, etc. are not analyzed and studied in a targeted manner, and no specific dual prism scanning technology solutions for these common scanning modes have been given. Summary of the invention

[0004] The technical problem to be solved by the present invention is to overcome the shortcomings of the existing dual-axis rotating two-dimensional scanning system, which is relatively complex in structure and large in size, and is not conducive to the small-scale integrated integration of the optoelectronic system. A portable laser radar two-dimensional scanning system and a scanning method based on a single-axis rotating double prism are provided, which can enable the outgoing light beam to perform various mode spatial scanning such as PPI, RHI, DBS, etc. within a large range, and is used to detect the spatial distribution of atmospheric physical quantities.

[0005] The technical solution adopted by the present invention to solve its technical problem is:

[0006] A portable laser radar two-dimensional scanning system based on a single-axis rotating double circular wedge prism includes two circular wedge prisms and two stepper motors. The system is characterized in that two identical circular wedge prisms RWP-1 and RWP-2 are coaxial and their standard surfaces (surfaces perpendicular to the edges) are placed back to back. Two identical stepper motors drive RWP-1 and RWP-2 to rotate independently around the axis respectively. A transmission mode is adopted between the stepper motor and the circular wedge prism in which a gear (active gear) fixed on the motor is meshed with a gear (passive gear) installed on the double circular wedge prisms. The rotation direction, angular velocity, etc. of the two stepper motors are controlled by a computer program.

[0007] The portable laser radar two-dimensional scanning system based on a single-axis rotating double circular wedge prism is characterized in that the material refractive index n of the double circular wedge prism is 2.03@532nm, the wedge angle α is 19.5°, and the values ​​of the refractive index n and the wedge angle α meet the critical total reflection condition at the last exit surface when the light beam is incident on the double circular wedge prism along the axis direction. By controlling the stepping motor to change the rotation angle of the double circular prism through a computer, when the light beam is incident on the double circular wedge prism along the axis direction, the exit light beam can be arbitrarily pointed within the range of 0-360° azimuth angle and 0-70.5° zenith angle, and the maximum zenith angle is equal to 90°-α.

[0008] The beam scanning using the above two-dimensional scanning system is achieved by the following method:

[0009] 1. According to the set beam scanning mode, determine the functional relationship between the zenith angle θ and the azimuth angle β of the outgoing beam and the time t, that is, θ(t) and β(t), and continuously and uniformly sample the time t over a long time interval to obtain t i , i = 1, 2, 3, ..., we can further get θ(t i ) and β(t i );

[0010] 2. Substitute the values ​​of n and α into θ p =α-arcsin(sinα / n), then n, α, θ p and θ(t i ) into the following formula to obtain cosθ i2 (t i )

[0011]

[0012] 3. Substitute n, α, and cosθ i2 (t i ),θ(t i ) and β(t i ) into the following formula, we get β2(t i )

[0013]

[0014] 4. Set n, α, and θ p ,θ(t i ) and β2(t i ) into the following formula, we get β1(t i )

[0015]

[0016] 5. Through steps 1 to 4, introduce different t i , and obtain a series of (t i , β1(t i )) and (t i , β2(t i )) data pairs, and then perform linear fitting or quadratic fitting according to the characteristics of the data, and determine which fitting method to use based on the degree to which the fitting correlation coefficient is close to 1;

[0017] If linear fitting is used, the fitting equation is β1=β 10 +ω1t,β2=β 20 +ω2t, thereby determining the initial angular displacement β of the circular wedge prism RWP-1 10 and the angular velocity ω1, and the initial angular displacement β of the circular wedge prism RWP-2 20 and the angular velocity ω2;

[0018] If quadratic fitting is used, the fitting equation is β1=β 10 +ω 10 t+γ1t 2 / 2, β2 = β 20 +ω 20 t+γ2t 2 / 2, thus determining the initial angular displacement β of the circular wedge prism RWP-1 10 , initial rotation angular velocity ω 10 , angular acceleration γ1, and the initial angular displacement β of the circular wedge prism RWP-2 20 , initial rotation angular velocity ω 20 , angular acceleration γ2;

[0019] 6. The rotation parameters of the double circular wedge prisms RWP-1 and RWP-2 obtained in step 5 are combined with the tooth ratio m of the active and passive gears to obtain the corresponding rotation parameters of the stepper motor 1 and the stepper motor 2. The controllers of the two stepper motors are then controlled by a computer to make the two stepper motors work according to the obtained rotation parameters.

[0020] Principle of the Invention

[0021] like Figure 1As shown, two identical circular wedge prisms (RWP) are coaxial and the standard faces (faces perpendicular to the edges) are placed back to back. They can be rotated independently around the axis by a stepper motor. Light beam 1 is incident from the inclined surface of RWP-1 along the axis, and after being transmitted through the standard surface of RWP-1, it is incident on the standard surface of RWP-2, and finally emerges from the inclined surface of RWP-2. By controlling the rotation angle β1 of RWP-1 and the rotation angle β2 of RWP-2, the zenith angle θ and azimuth angle β of the outgoing light beam 2' can be arbitrarily changed within a certain range.

[0022] The forward analytical solution of the outgoing state is obtained from the incident state. That is, the rotation angles β1 and β2 of RWP-1 and RWP-2 are known, and the zenith angle θ and azimuth angle β of the outgoing beam are obtained. The process is as follows:

[0023] Assume that the two inclined surfaces of RWP-1 and RWP-2 are parallel in the initial state, the wedge angles are both α, the refractive index is both n, and the symmetry plane of the double circular wedge prism is set to the rotation zero position, that is, β1=β2=0, as Figure 2 (a) According to the law of refraction, the propagation trajectory of the light beam from the rotating double circular wedge prism to the exit is obtained, as shown in Figure 2 (b) as shown; Figure 2 (b) The beam propagation trajectory is analyzed in depth, and a simplified model of the beam propagation trajectory in the rotating double circular wedge prism is obtained, as shown in Figure 2 (c) as shown. Figure 2 In (c), OO' is the coaxial line, OP is the normal line of the inclined surface of RWP-2. When RWP-2 rotates around the axis, OP will also rotate around the axis and form a conical surface with OP as the generatrix in the figure. AO is the light beam incident on the inclined surface of RWP-2, and BO is the refracted light beam of the inclined surface of RWP-2.

[0024] like Figure 2 As shown in (b), assuming that the circular wedge prism RWP-1 is rotated by an angle β1 around the coaxial line, the light beam 1 is incident along the axis from the inclined surface of RWP-1, and the incident angle θ i1 =α. Applying the law of refraction and geometric relationships, we can obtain the deflection angle θ of the refracted beam 1' on the RWP-1 inclined surface: p for:

[0025] θ p =θ i1 -θ r1 =α-arcsin(sinα / n) (1)

[0026] Where: θ r1is the refraction angle of the light beam. It should be noted that at this time, the angle between the incident light beam, the refracted light beam and the plane where the normal lies and the set rotation direction of zero degrees is β1. Next, the refracted light beam 1' propagates from the inside of RWP-1 to the inside of RWP-2, passing through the parallel air gap between the two circular wedge prisms. It is easy to conclude that in this process, the direction of the light beam 1' remains unchanged after two refractions, and there is only a small translation related to the thickness of the air gap. Assuming that the thickness of the air gap is very small, that is, the two circular wedge prisms are very close, the small translation can be ignored, and the light beam entering the inside of RWP-2 is still represented by 1', that is Figure 2 AO in (c).

[0027] like Figure 2 As shown in (c), assuming that the circular wedge prism RWP-2 is rotated by an angle of β2 around the coaxial line, the incident angle of the light beam 1' on the inclined surface of RWP-2 is θ i2 , the refraction angle is θ r2 , obviously there is sinθ r2 =nsinθ i2 In the figure, Using the cosine theorem in ΔAO'P and ΔAOP, we can obtain:

[0028] c 2 =(l·tanα) 2 +(l·tanθ p ) 2 -2l 2 tanαtanθ p ·cos(β2-β1) (2)

[0029] c 2 =(l / cosα) 2 +(l / cosθ p ) 2 -2l 2 / (cosαcosθ p )·cosθ i2 (3)

[0030] From equations (2) and (3), we can get:

[0031] cosθ i2 = cosθ p cosα+sinθ p sinαcos(β2-β1) (4)

[0032] Using the cosine theorem in ΔBO'P and ΔBOP, we can obtain:

[0033] a 2 =(ltanα) 2 +(ltanθ) 2-2l 2 tanαtanθ·cos(β2-β) (5)

[0034] a 2 =(l / cosα) 2 +(l / cosθ) 2 -2l 2 / (cosαcosθ)·cosθ r2 (6)

[0035] From equations (5) and (6), we can get:

[0036] cosθ r2 =cosθcosα+sinθsinαcos(β2-β) (7)

[0037] Using the sine theorem in ΔBOP and ΔAOP, we can obtain:

[0038] a / sinθ r2 =(l / cosθ) / sin∠APO (8)

[0039] c / sinθ i2 =(l / cosθ p ) / sin∠APO (9)

[0040] From equations (8) and (9), we can get:

[0041]

[0042] Using the sine theorem in ΔBOP and ΔBOA, we can obtain:

[0043] a / sinθ r2 =(l / cosα) / sin∠ABO (11)

[0044] (ac) / sin(θ r2 -θ i2 )=(l / cosθ p ) / sin∠ABO (12)

[0045] From equations (11) and (12), we can get:

[0046]

[0047] Combining (10) and (13) and simplifying them, we can get:

[0048] cosθ=ncosθ p +(cosθ r2 -ncosθ i2 )cosα (14)

[0049] According to sinθ r2 =nsinθ i2 And θ r2 ≤90°, we can get:

[0050]

[0051] Substituting equation (15) into equation (14) and equation (7), we obtain

[0052]

[0053]

[0054] in:

[0055]

[0056] Substituting equations (1) and (4) into equation (16), we can obtain the functional relationship of θ(β1, β2); and substituting equations (1), (4) and (16) into equation (17), we can obtain the functional relationship of β(β1, β2).

[0057] Conversely, the reverse analytical solution of the incident state is obtained from the outgoing state, that is, the zenith angle θ and azimuth angle β of the outgoing beam are known, and the rotation angles β1 and β2 of RWP-1 and RWP-2 are obtained. The process is as follows:

[0058] For simplicity, it is assumed here that 0≤β2-β1≤180°. From equation (16), we can obtain:

[0059]

[0060] From (17), we get:

[0061]

[0062] Combining equations (4) and (18), we can obtain:

[0063]

[0064] Substituting equations (1) and (18) into equation (19), we can obtain the functional relationship of β2(θ, β); and substituting equations (1) and (19) into equation (20), we can obtain the functional relationship of β1(θ, β).

[0065] It is easy to find from formula (4):

[0066] cos(θ p +a)≤cosθ i2 ≤cos(θ p -α) (21)

[0067] When β2-β1=π, θ i2 Take the maximum value θ p +α. At this time, on the inclined surface of the circular wedge prism RWP-2, in order for the light beam 1' to propagate forward, the following conditions must be met:

[0068] sin(θ p +α)≤1 / n (22)

[0069] When the above equation takes "=", the beam 1' is in the critical full emission state, and the zenith angle θ of the outgoing beam takes the maximum value of 90°-α. In the critical full emission state, substitute equation (1) into equation (22) to simplify it:

[0070] 4n 2 sin 3 α+4(n 2 -1)sin 2 α-3sinα-1=0 (23)

[0071] According to formula (23), the relationship curve of the prism wedge angle α and the refractive index n when the critical total reflection condition is met can be obtained, as shown in Figure 3 shown.

[0072] On the other hand, when the prism wedge angle α and the refractive index n meet the critical total reflection condition, the range of the zenith angle of the outgoing light beam 2' will reach the maximum, that is, [0, 90°-α]. Therefore, in order to make the scanning range of the outgoing light beam as large as possible, it is necessary to make the value of α as small as possible while meeting the critical total reflection condition. Figure 3 It can be seen that in order to make the α value small, the wedge prism should be made of glass material with a high refractive index as much as possible. Here, LASF35, a high refractive index lanthanide optical glass from Schott, is selected. Assuming that the working wavelength of the portable wind laser radar is 532nm, the refractive index n of LASF35 at this wavelength is 2.03. Therefore, according to Figure 3 Here, the refractive index n of the circular wedge prism material is selected to be 2.03, and the wedge angle α is 19.5°. At this time, the zenith angle scanning range of the outgoing light beam 2' is [0, 70.5°], that is, the elevation angle scanning range is [19.5°, 90°].

[0073] From equations (16) and (17), a three-dimensional graph can be drawn showing the zenith angle θ and azimuth angle β of the outgoing light beam as the rotation angles β1 and β2 of the double circular wedge prisms RWP-1 and RWP-2 vary from 0 to 360° (see Figure 4 (a) and Figure 4 (b)). The corresponding contour maps are as follows: Figure 4 (c) and Figure 4 (d) is shown. Figure 4It can be seen that when RWP-1 and RWP-2 are rotated one circle, that is, β1 and β2 change from 0 to 360°, the output beam can be arbitrarily changed in the range of 0 to 70.5° zenith angle and 0 to 360° azimuth angle. If the limiting condition 0≤β2-β1≤180° is added, it will not affect the output beam to change arbitrarily in the range of 0 to 70.5° zenith angle and 0 to 360° azimuth angle.

[0074] From equations (19) and (20), we can draw the relationship curves of the rotation angles β1 and β2 of the double circular wedge prisms RWP-1 and RWP-2 with the zenith angle θ and azimuth angle β of the outgoing beam. In the PPI scanning mode, when the outgoing beam adopts different scanning zenith angles θ, the relationship between the rotation angles β1 and β2 of the double circular wedge prisms RWP-1 and RWP-2 and the azimuth angle β of the outgoing beam can be seen in Figure 5 .from Figure 5 It can be seen that once the value of θ is determined, there is a linear relationship between β and β1, and between β and β2, and φ1 = β2-β1 is a constant; as θ increases, φ1 also gradually increases. When θ is 45°, 60°, and 70°, φ1 is 120.94°, 156.47°, and 178.87°, respectively. In the RHI scanning mode, when the outgoing beam takes different scanning azimuth angles β, the relationship curve of the rotation angles β1 and β2 of the double circular wedge prisms RWP-1 and RWP-2 with the change of the zenith angle θ of the outgoing beam is shown in Figure 1. Figure 6 .from Figure 6 It can be seen that: once the β value is determined, there is an approximately linear relationship between θ and β1, and between θ and β2; changes in the β value will not affect the current state of the curve, but will only cause the overall curve to shift up and down; when β increases from 90° to 180°, the two curves are also raised by 90° as a whole, and the overall shift of the curve is equal to the change in β, and when θ = 0°, the initial values ​​of β1 and β2 are equal, both β + 90°.

[0075] Due to the adoption of the above technical scheme, compared with the existing two-dimensional scanning system, the advantages and positive effects of the present invention are as follows: 1. A scanning system based on a uniaxially rotating double circular wedge prism is adopted, which has a compact structure and is easy to use; 2. The wedge angle α and the refractive index n of the wedge prism are optimized and designed. Under the condition of critical total reflection of the last exit surface, the refractive index n is taken as large as possible and the wedge angle α is taken as small as possible. By rotating the double circular wedge prism, the exit light beam can be arbitrarily pointed in a large range of 0 to 360° azimuth angle and 0 to 90°-α zenith angle; 3. The control of the rotation speed and direction of the double circular wedge prism can enable the exit light beam to perform accurate scanning in different modes such as PPI, RHI, DBS, etc. BRIEF DESCRIPTION OF THE DRAWINGS

[0076] Figure 1 It is a structural block diagram of the present invention.

[0077] Figure 1 1. Round wedge prism RWP-1, 2. Round wedge prism RWP-2, 3. Stepper motor 1, 4. Stepper motor 2, 5. Computer.

[0078] Figure 2 It is the light beam propagation trajectory and its simplified model of the present invention.

[0079] Figure 3 It is a curve showing the change of the prism wedge angle with the refractive index under the critical total reflection condition of the present invention.

[0080] Figure 4 This is the relationship between the zenith angle θ and the azimuth angle β of the outgoing light beam of the present invention and the rotation angles β1 and β2 of RWP-1 and RWP-2.

[0081] Figure 5 It is the relationship between the rotation angles β1 and β2 of RWP-1 and RWP-2 and the azimuth angle β at different zenith angles θ in the PPI scanning mode of the present invention.

[0082] Figure 6 It is the relationship between the rotation angles β1 and β2 of RWP-1 and RWP-2 and the zenith angle θ at different azimuth angles β in the RHI scanning mode of the present invention. DETAILED DESCRIPTION

[0083] The present invention is further described below in conjunction with the embodiments and drawings, but the protection scope of the present invention shall not be limited thereby. The specific implementation method of the present invention is as follows:

[0084] The structural block diagram of the present invention is as follows: Figure 1 shown. Figure 1 The middle light beam 1' is incident on the circular wedge prism RWP-1(1) along the axis, passes through the parallel air gap between the two circular wedge prisms, and is emitted from the circular wedge prism RWP-2(2). The stepper motor 1(3) and the stepper motor 2(4) are controlled by the computer (5) to rotate RWP-1(1) and RWP-2(2) respectively, so that the zenith angle θ and azimuth angle β of the emitted light beam 2' can be arbitrarily changed within a certain range. The wedge angle α of the two circular wedge prisms is 19.5° and the refractive index is 2.03.

[0085] a) In PPI scanning mode, by Figure 5 According to the analysis, the angle β2-β1 is calculated according to the set scanning zenith angle θ. First, the stepper motor 2 driving the circular wedge prism RWP-2 controls the RWP-2 to rotate at an angle β2-β1; then the two stepper motors are rotated at the same speed. The angular velocity ω is determined by the scanning period T, that is, ω=2π / (Tm), where m is the gear ratio of the main and driven gears.

[0086] b) In RHI scanning mode, by Figure 6 According to the analysis, when the positioning accuracy requirement is not very high, according to the set scanning azimuth angle β, first let the two stepper motors drive the circular wedge prisms RWP-1 and RWP-2 to rotate β+90°; then let the stepper motor 1 driving the circular wedge prism RWP-1 and the stepper motor 2 driving the circular wedge prism RWP-2 rotate at an angular velocity ω1=k1θ max / (Tm) and ω2=k2θ max / (Tm) uniform rotation, where θ max is the set maximum scanning zenith angle, T is the scanning period, k1 and k2 are Figure 6 The linear fitting slope of the middle curve θ-β1 and θ-β2. Here, when α=19.5°, n=2.03, k1=-1.27, k2=1.29, the correlation coefficient of the fitting is R 2 ω1 is a negative value, indicating that the motor driving the circular wedge prism RWP-1 rotates in the opposite direction. When the positioning accuracy is very high, it is necessary to use quadratic fitting for the curves θ-β1 and θ-β2. The correlation coefficient R 2 They are 0.99999 and 0.99995 respectively. At this time, it is necessary to control the two stepper motors to rotate at a uniform speed.

Claims

1. A portable laser radar two-dimensional scanning system based on a single-axis rotating double circular wedge prism, comprising two circular wedge prisms and two stepping motors, characterized in that Two identical circular wedge prisms RWP-1 and RWP-2 are coaxial and placed with their standard surfaces facing each other. Two identical stepper motors drive RWP-1 and RWP-2 to rotate independently around their axes. The stepper motor and the circular wedge prisms are connected by a gear fixed on the motor and a gear mounted on the double circular wedge prisms. The rotation direction and angular velocity of the two stepper motors are controlled by a computer program. The portable laser radar two-dimensional scanning system based on a uniaxially rotating double circular wedge prism is characterized in that the material refractive index n of the double circular wedge prism is 2.03 at a wavelength of 532nm, and the wedge angle α is 19.5°, and the values ​​of the refractive index n and the wedge angle α meet the critical total reflection condition at the last exit surface when the light beam is incident on the double circular wedge prism along the axial direction; the rotation angle of the double circular prism is changed by controlling the stepping motor by a computer, so that when the light beam is incident on the double circular wedge prism along the axial direction, the exit light beam can be arbitrarily pointed within the range of 0-360° azimuth angle and 0-70.5° zenith angle, and the maximum zenith angle is equal to 90°-α; The beam scanning using the above two-dimensional scanning system is achieved by the following method: ①, according to the set beam scanning mode, determine the functional relationship between the zenith angle θ and the azimuth angle β of the outgoing beam and the time t, that is, θ(t) and β(t), and continuously and uniformly sample the time t over a long time interval to obtain t i , i = 1, 2, 3, ..., we can further get θ(t i ) and β(t i ); ②Substitute the values ​​of n and α into θ p =α-arcsin(sinα / n), then n, α, θ p and θ(t i ) into the following formula to obtain cosθ i2 (t i ) ③、n、α、cosθ i2 (t i ),θ(t i ) and β(t i ) into the following formula, we get β2(t i ) ④、n、α、θ p ,θ(t i ) and β2(t i ) into the following formula, we get β1(t i ) ⑤. Through steps 1 to 4, introduce different t i , and obtain a series of (t i , β1(t i )) and (t i , β2(t i )) data pairs, and then perform linear fitting or quadratic fitting according to the characteristics of the data, and determine which fitting method to use based on the degree to which the fitting correlation coefficient is close to 1; If linear fitting is used, the fitting equation is β1=β 10 +ω1t,β2=β 20 +ω2t, thereby determining the initial angular displacement β of the circular wedge prism RWP-1 10 and the angular velocity ω1, and the initial angular displacement β of the circular wedge prism RWP-2 20 and the angular velocity ω2; If quadratic fitting is used, the fitting equation is β1=β 10 +ω 10 t+γ1t 2 / 2, β2 = β 20 +ω 20 t+γ2t 2 / 2, thus determining the initial angular displacement β of the circular wedge prism RWP-1 10 , initial rotation angular velocity ω 10 , angular acceleration γ1, and the initial angular displacement β of the circular wedge prism RWP-2 20 , initial rotation angular velocity ω 20 , angular acceleration γ2; ⑥. Combine the rotation parameters of the double circular wedge prisms RWP-1 and RWP-2 obtained in step 5 with the tooth ratio m of the active and passive gears to obtain the corresponding rotation parameters of stepper motor 1 and stepper motor 2. Then use a computer to control the controllers of the two stepper motors so that the two stepper motors work according to the obtained rotation parameters.

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