Issue #6/2017

A Device Constructing Local Vertical With a Panoramic Mirror-Lens System

**A.V.Melnikov, V.A.Solomatin**A Device Constructing Local Vertical With a Panoramic Mirror-Lens System

The basic scheme and parameters of the static device for orientation on the Earth (the device constructing local vertical) of the space vehicle (SC) on the basis of the microbolometer matrix using a panoramic mirror lens (panoramic annular lens - PAL-lens) are considered.

Теги: device for local vertical constructing panoramic mirror-lens system панорамный зеркально-линзовый объектив построитель местной вертикали

The devices constructing local vertical (DLV), working by contrast between the surface of the planet and the outer cosmic space in the infrared spectrum are widely used in the spacecraft control systems. In the widespread cross-type DLV, an optical-mechanical scanning, providing a "section" of the Earth’s horizon by a narrow instantaneous angular field [1] is performed. An intention to reduce the DLV energy consumption, decrease their overall dimensions and weight, enhance reliability leads to engineering solutions that exclude optical-mechanical scanning. One of such solution is the use of panoramic optical systems covering the entire horizon with an angular field, in combination with multiple-unit radiation detector arrays (photodetector arrays – PHAs), whereby various methods of panoramic image analysis and its preliminary processing (delay, interpolation, integration, scaling, filtering) can be used.

PANORAMIC OPTICAL SYSTEM

Among the possible ways of panoramic optical systems constructing covering the whole horizon [2], mirror-lens structures known as PAL-lenses (panoramic annular lens) are the most promising for use in DLV. The lens of the PAL lens-based positioning mechanism was successfully applied in the SEASIS system (The SEDS Earth Atmosphere and Space Imaging System) for the SEDSAT‑1 micro-satellite [3].The original version of the PAL-lens was designed at MIIGAiK [4]. Compared to the known similar optical systems, the lens designed at MIIGAiK does not have aspheric surfaces and has a large angular field. The lens is a mono-structure bounded by four surfaces: the first surface is a spherical refracting surface, the second and third surfaces are spherical reflecting, and the fourth is a plane refractive one. The image created by the PAL-lens is inside the glass, so an additional transmitting lens transferring the image to the plane beyond the lens (figure 1) is required.

PAL-lenses create an annular space image corresponding to the cylindrical projection (fig. 2). The width of the ring corresponds to α angle, blind spot to 2β angle.

The use of an annular panoramic lens will make it possible to obtain an image of the entire horizon of the Earth with smaller dimensions and mass compared to the known wide-angle fish-eye type lenses and without spatial distortions inherent in such systems in azimuth angle transformation in the range from to.

A functional diagram of an optical DLV unit using a PAL-lens is shown in fig. 3. To provide a measurement range of the deviation angle from the vertical ±3°, taking into account the altitude change and the device installation error, it is required to build a "space-to-Earth" section on the PHA with an angular dimension of around 10°, which includes margin for the aberration of the optical system, tolerance on a startup accuracy of the SC and orbit ellipticity [5]. With the indicated parameters and flight altitude equal to 650 km, the angular dimension of the Earth will constitute 130°, therefore, the maximum value of the lens angular field shall be 140°, and the minimum 120°.

To perform the calculations, a microbolometer matrix designed at the A. V. Rzhanov Institute of Semiconductor Physics is accepted as a radiation detector with the following specifications [6]:

• spectral sensitivity range – 8–14 µm;

• format 320 Ч 240;

• elements step of the receivers matrix –51 µm;

• temperature resolution mK.

Let us assume the radius of the outer ring of the Earth image equal to mm (half the length of the matrix’s side), the angles , . The radius of the inner ring is defined as [7]:

.

At ,mm [7], and the angular size of the pixel is 33′. The 18 elements of the matrix correspond to the width of the annular image in an angular measure of 10°. Since half of them are occupied by the image of the cosmos, the deflection angle is defined by 220 lines of 240.

THE DEFLECTION ANGLE ALGORITHM

The algorithm of signals transformation from a PHA is designed for preliminary frame processing, signals correction and an output information computing – roll and pitch deflection angles of the spacecraft.

As a result of each frame processing received from electro optical probing payload, the following tasks are solved:

1. The matrix is calibrated against a uniform background. The transmission function for each pixel is determined and the calibration factor Kij is stored in the memory of the calculator. Signals from pixels with insufficient sensitivity or those, whose noise exceeds the allowable value, are replaced by the averages of the surrounding pixels.

2. Interfering sources of radiation, sorted by the angular size, are detected and excluded (brought to the surrounding background level).

The matrix design allows performing temporary algorithmic information deactivation from response elements, the angular field of which catches the Sun, the Moon or elements of the spacecraft body (for example, antennas or solar cell batteries).

3. The deviation of the SC axis from the direction to the center of the Earth is determined, that is, the local vertical is constructed.

The problem of determining the deviation of the SC axis from the direction to the center of the Earth can be solved by the coordinates averaging method at which the following operations are performed.

• In the frame of M Ч N dimension (where N is the number of lines in the frame, M is the number of resolution elements in one line), the coordinates of the resolution elements corresponding to the contour points of the observed figure of the Earth j, I1 and j, I2, are indicted (fig. 4).

• The coordinates and are found wherein the signal level will be equal to the values of Uп1 = 1/3 and Uп2 = 2/3 of the maximum Umax signal at the minimum brightness of the Earth, respectively, and the assumption of linearity dependence of the signal amplitude on the sighting angle (Fig.5).

• Interpolation is performed, allowing to reduce the positioning error of and coordinates and resulting from pixels discreteness.

For linear interpolation:

,

θi – angular coordinates of the pixel whose signal interrogation exceeds the value Uп1, Δ – pixel angular pitch of the radiation receiver, Ui – 1, Ui – signals from pixels along the line with the coordinates i‑1, i respectively. Linear interpolation allows determining image coordinate in the plane of a multi-element radiation receiver with a maximum error of 1/10 to 1/100 of the element size [8].

• Brightness correction is performed, since the Umax signal can vary four times depending on the Earth brightness [9]. The steepness of the response region between the two thresholds allows determining the value of the brightness correction. Then the refined value of the first threshold corresponds to the angular coordinate of the space-to-Earth boundary along the column j (fig. 6):

,

where П is the correction that is entered in the signal processing algorithm from matrix elements, depending on the Δ1 value. If we consider the region between the two signal thresholds to be linear, then we shall accept

П = θп2 – θп1.

• The averaged value of the horizon side position is determined as:

,

where N1 and N2 are the line numbers corresponding to the boundaries of the Earth by the column j

Since the angular field of the device is greater than the angular dimension of the Earth, the signal from the space is always reliably fixed as the minimum level with respect to which the thresholds Uп1 и Uп2 are set. The value θcp2 for the second side of the horizon is determined in a similar way.

• The deviation of the SC axis from the direction to the center of the Earth (j0, i0) is computed as

.

ACCURACY EVALUATION

To determine competitive performance of the DLV proposed engineering solution, it is necessary to evaluate the accuracy of constructing the local vertical. Let us consider the most essential components of measurement error: fluctuation (noise), instrumental and systematic.

Let us determine the fluctuation error. The steepness of the "space-to-Earth" energy front (Figure 5) is defined as

.

Let us represent the k value as the radiation flux ratio from the Earth Фср = Вср = Авх = Ωпр with an average brightness in a given range of Bср to the value of the space-to-Earth transition, composing an angle θ by the blurring of the Earth’s atmosphere:

,

where Авх – is the area of the entrance pupil, Ωпр – is the angular dimension of the matrix element

Noise component of the signal:

,

where η is the signal-to-noise ratio.

Then the root-mean-square value of the fluctuation error will be equal to:

.

Let us represent the signal-to-noise ratio as the radiant flux ratio at the receiver site when working on a perfect radiator Ф to the minimum resolved flux Фш:

where σ is the Stefan-Boltzmann constant, T1 is the temperature of the perfect radiator, τoпт is the transmittance factor of the optical system, ρ is the factor considering reflection losses, T2 is the mean radiant temperature of the Earth.

For a system with an entrance pupil diameter of D = 4 mm, the signal-to-noise ratio will be η = 360. Then at T1 = 300 К, τoпт = 0,8 [10], ρ = 0,97, T2 = 250 K [1], , θ = 3° the root-mean-square value of the fluctuation error will be σш = 0,5’.

The alignment error of the device optical part and the geometric error in determining the angle corresponding to the "Earth-to-space" boundary (of the horizon angle) will be considered as components of the instrumental error. Alignment of the device optical part is provided by sensitive imitation equipment with a maximum error of Δю = 1′.

The error limit in calculating the angular coordinates of the horizon θпi can be reduced by interpolation to a value of not more than 0.5′. Then, considering the averaging over the matrix lines, this error reduces by a factor of and be equal to:

.

Systematic error source of the device is the difference between the working source – the Earth and an ideal even sphere.

The magnitude of this error does not depend on the type of device and is determined only by the geometric parameters of the planet and the orbit of the ESV and the variations in the brightness of the Earth. The maximum value of this error occurs in the meridian direction at a latitude of the sub-satellite point of L0 = 45 ° and for the flight altitude of H = 650 km varies within ±9′ and can be completely compensated by an algorithmic method [11].

The residual root-mean-square error due to latitudinal variation of brightness does not exceed σя = 1′.

Due to random meteorological factors the error depends on the brightness of the Earth’s IR horizon and at an average brightness equals to 3′ [11]. The average size of the inhomogeneity sections of the atmosphere for the 8–14-µm range is 600 km (average cloud cover size), then at a flight altitude of 650 km, about 11 sections of the atmospheric inhomogeneity are accounted for the aimed horizon area, as a result of averaging this value decreases to 0.9′:

.

In the window region (8–12 µm), there is a significant variation in both the brightness and the height of the radiating atmosphere in the range from the firm horizon to the upper layers of the troposphere. The root-mean-square error of the radiating atmosphere height is σh = 2.5 km. In an angular measure, considering averaging, it will be σh = 0,9′ [11].

Assuming the error Δю being a systematic, the errors of σш, σг, σм, σя – random and independent with normal distribution, the error σh = 0,9′ – random with uniform distribution, we will get the net inaccuracy formula of the form

.

When substituting numerical values to this formula, we get Δсум = 5,7′.

CONCLUSION

The developed schematic diagram of the DLV based on the catadioptric lens (the panoramic ring PAL lens) and the microbolometer matrix excludes the optical-mechanical scanning, simplifies the design, increases the service life of the device, reduces its power consumption, and also simplifies the positioning of the DLV on the spacecraft body. The proposed signal conversion algorithm allows performing brightness correction of the signal in determining the angle of deviation of the spacecraft axis from the vertical.

The designed accuracy of the DLV applying the proposed engineering solution is characterized by the limiting error of not more than 5.7′ in the range of displacement angles ±3° at the altitude of the spacecraft flight up to 1 000 km, which is comparable to the secant type DLV (with optical-mechanical scanning).

Advantages of the proposed engineering solution are particularly topical for small spacecraft with limited energy consumption and a long service life.

PANORAMIC OPTICAL SYSTEM

Among the possible ways of panoramic optical systems constructing covering the whole horizon [2], mirror-lens structures known as PAL-lenses (panoramic annular lens) are the most promising for use in DLV. The lens of the PAL lens-based positioning mechanism was successfully applied in the SEASIS system (The SEDS Earth Atmosphere and Space Imaging System) for the SEDSAT‑1 micro-satellite [3].The original version of the PAL-lens was designed at MIIGAiK [4]. Compared to the known similar optical systems, the lens designed at MIIGAiK does not have aspheric surfaces and has a large angular field. The lens is a mono-structure bounded by four surfaces: the first surface is a spherical refracting surface, the second and third surfaces are spherical reflecting, and the fourth is a plane refractive one. The image created by the PAL-lens is inside the glass, so an additional transmitting lens transferring the image to the plane beyond the lens (figure 1) is required.

PAL-lenses create an annular space image corresponding to the cylindrical projection (fig. 2). The width of the ring corresponds to α angle, blind spot to 2β angle.

The use of an annular panoramic lens will make it possible to obtain an image of the entire horizon of the Earth with smaller dimensions and mass compared to the known wide-angle fish-eye type lenses and without spatial distortions inherent in such systems in azimuth angle transformation in the range from to.

A functional diagram of an optical DLV unit using a PAL-lens is shown in fig. 3. To provide a measurement range of the deviation angle from the vertical ±3°, taking into account the altitude change and the device installation error, it is required to build a "space-to-Earth" section on the PHA with an angular dimension of around 10°, which includes margin for the aberration of the optical system, tolerance on a startup accuracy of the SC and orbit ellipticity [5]. With the indicated parameters and flight altitude equal to 650 km, the angular dimension of the Earth will constitute 130°, therefore, the maximum value of the lens angular field shall be 140°, and the minimum 120°.

To perform the calculations, a microbolometer matrix designed at the A. V. Rzhanov Institute of Semiconductor Physics is accepted as a radiation detector with the following specifications [6]:

• spectral sensitivity range – 8–14 µm;

• format 320 Ч 240;

• elements step of the receivers matrix –51 µm;

• temperature resolution mK.

Let us assume the radius of the outer ring of the Earth image equal to mm (half the length of the matrix’s side), the angles , . The radius of the inner ring is defined as [7]:

.

At ,mm [7], and the angular size of the pixel is 33′. The 18 elements of the matrix correspond to the width of the annular image in an angular measure of 10°. Since half of them are occupied by the image of the cosmos, the deflection angle is defined by 220 lines of 240.

THE DEFLECTION ANGLE ALGORITHM

The algorithm of signals transformation from a PHA is designed for preliminary frame processing, signals correction and an output information computing – roll and pitch deflection angles of the spacecraft.

As a result of each frame processing received from electro optical probing payload, the following tasks are solved:

1. The matrix is calibrated against a uniform background. The transmission function for each pixel is determined and the calibration factor Kij is stored in the memory of the calculator. Signals from pixels with insufficient sensitivity or those, whose noise exceeds the allowable value, are replaced by the averages of the surrounding pixels.

2. Interfering sources of radiation, sorted by the angular size, are detected and excluded (brought to the surrounding background level).

The matrix design allows performing temporary algorithmic information deactivation from response elements, the angular field of which catches the Sun, the Moon or elements of the spacecraft body (for example, antennas or solar cell batteries).

3. The deviation of the SC axis from the direction to the center of the Earth is determined, that is, the local vertical is constructed.

The problem of determining the deviation of the SC axis from the direction to the center of the Earth can be solved by the coordinates averaging method at which the following operations are performed.

• In the frame of M Ч N dimension (where N is the number of lines in the frame, M is the number of resolution elements in one line), the coordinates of the resolution elements corresponding to the contour points of the observed figure of the Earth j, I1 and j, I2, are indicted (fig. 4).

• The coordinates and are found wherein the signal level will be equal to the values of Uп1 = 1/3 and Uп2 = 2/3 of the maximum Umax signal at the minimum brightness of the Earth, respectively, and the assumption of linearity dependence of the signal amplitude on the sighting angle (Fig.5).

• Interpolation is performed, allowing to reduce the positioning error of and coordinates and resulting from pixels discreteness.

For linear interpolation:

,

θi – angular coordinates of the pixel whose signal interrogation exceeds the value Uп1, Δ – pixel angular pitch of the radiation receiver, Ui – 1, Ui – signals from pixels along the line with the coordinates i‑1, i respectively. Linear interpolation allows determining image coordinate in the plane of a multi-element radiation receiver with a maximum error of 1/10 to 1/100 of the element size [8].

• Brightness correction is performed, since the Umax signal can vary four times depending on the Earth brightness [9]. The steepness of the response region between the two thresholds allows determining the value of the brightness correction. Then the refined value of the first threshold corresponds to the angular coordinate of the space-to-Earth boundary along the column j (fig. 6):

,

where П is the correction that is entered in the signal processing algorithm from matrix elements, depending on the Δ1 value. If we consider the region between the two signal thresholds to be linear, then we shall accept

П = θп2 – θп1.

• The averaged value of the horizon side position is determined as:

,

where N1 and N2 are the line numbers corresponding to the boundaries of the Earth by the column j

Since the angular field of the device is greater than the angular dimension of the Earth, the signal from the space is always reliably fixed as the minimum level with respect to which the thresholds Uп1 и Uп2 are set. The value θcp2 for the second side of the horizon is determined in a similar way.

• The deviation of the SC axis from the direction to the center of the Earth (j0, i0) is computed as

.

ACCURACY EVALUATION

To determine competitive performance of the DLV proposed engineering solution, it is necessary to evaluate the accuracy of constructing the local vertical. Let us consider the most essential components of measurement error: fluctuation (noise), instrumental and systematic.

Let us determine the fluctuation error. The steepness of the "space-to-Earth" energy front (Figure 5) is defined as

.

Let us represent the k value as the radiation flux ratio from the Earth Фср = Вср = Авх = Ωпр with an average brightness in a given range of Bср to the value of the space-to-Earth transition, composing an angle θ by the blurring of the Earth’s atmosphere:

,

where Авх – is the area of the entrance pupil, Ωпр – is the angular dimension of the matrix element

Noise component of the signal:

,

where η is the signal-to-noise ratio.

Then the root-mean-square value of the fluctuation error will be equal to:

.

Let us represent the signal-to-noise ratio as the radiant flux ratio at the receiver site when working on a perfect radiator Ф to the minimum resolved flux Фш:

where σ is the Stefan-Boltzmann constant, T1 is the temperature of the perfect radiator, τoпт is the transmittance factor of the optical system, ρ is the factor considering reflection losses, T2 is the mean radiant temperature of the Earth.

For a system with an entrance pupil diameter of D = 4 mm, the signal-to-noise ratio will be η = 360. Then at T1 = 300 К, τoпт = 0,8 [10], ρ = 0,97, T2 = 250 K [1], , θ = 3° the root-mean-square value of the fluctuation error will be σш = 0,5’.

The alignment error of the device optical part and the geometric error in determining the angle corresponding to the "Earth-to-space" boundary (of the horizon angle) will be considered as components of the instrumental error. Alignment of the device optical part is provided by sensitive imitation equipment with a maximum error of Δю = 1′.

The error limit in calculating the angular coordinates of the horizon θпi can be reduced by interpolation to a value of not more than 0.5′. Then, considering the averaging over the matrix lines, this error reduces by a factor of and be equal to:

.

Systematic error source of the device is the difference between the working source – the Earth and an ideal even sphere.

The magnitude of this error does not depend on the type of device and is determined only by the geometric parameters of the planet and the orbit of the ESV and the variations in the brightness of the Earth. The maximum value of this error occurs in the meridian direction at a latitude of the sub-satellite point of L0 = 45 ° and for the flight altitude of H = 650 km varies within ±9′ and can be completely compensated by an algorithmic method [11].

The residual root-mean-square error due to latitudinal variation of brightness does not exceed σя = 1′.

Due to random meteorological factors the error depends on the brightness of the Earth’s IR horizon and at an average brightness equals to 3′ [11]. The average size of the inhomogeneity sections of the atmosphere for the 8–14-µm range is 600 km (average cloud cover size), then at a flight altitude of 650 km, about 11 sections of the atmospheric inhomogeneity are accounted for the aimed horizon area, as a result of averaging this value decreases to 0.9′:

.

In the window region (8–12 µm), there is a significant variation in both the brightness and the height of the radiating atmosphere in the range from the firm horizon to the upper layers of the troposphere. The root-mean-square error of the radiating atmosphere height is σh = 2.5 km. In an angular measure, considering averaging, it will be σh = 0,9′ [11].

Assuming the error Δю being a systematic, the errors of σш, σг, σм, σя – random and independent with normal distribution, the error σh = 0,9′ – random with uniform distribution, we will get the net inaccuracy formula of the form

.

When substituting numerical values to this formula, we get Δсум = 5,7′.

CONCLUSION

The developed schematic diagram of the DLV based on the catadioptric lens (the panoramic ring PAL lens) and the microbolometer matrix excludes the optical-mechanical scanning, simplifies the design, increases the service life of the device, reduces its power consumption, and also simplifies the positioning of the DLV on the spacecraft body. The proposed signal conversion algorithm allows performing brightness correction of the signal in determining the angle of deviation of the spacecraft axis from the vertical.

The designed accuracy of the DLV applying the proposed engineering solution is characterized by the limiting error of not more than 5.7′ in the range of displacement angles ±3° at the altitude of the spacecraft flight up to 1 000 km, which is comparable to the secant type DLV (with optical-mechanical scanning).

Advantages of the proposed engineering solution are particularly topical for small spacecraft with limited energy consumption and a long service life.

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