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An intensified high dynamic star tracker (IHDST) is a photoelectric instrument and stably outputs three-axis attitude for a spacecraft at very high angular velocity. The IHDST uses an image intensifier to multiply the incident starlight. Thus, high sensitivity of the star detection is achieved under short exposure time such that extremely high dynamic performance is achieved. The IHDST differs from a traditional star tracker in terms of the imaging process. Therefore, we establish a quantum transfer model of IHDST based on stochastic process theory. By this model, the probability distribution of the output quantum number is obtained accurately. Then, we introduce two-dimensional Lorentz functions to describe the spatial spreading process of the IHDST. Considering the interaction of these two processes, a complete star imaging model of IHDST is provided. Using this model, the centroiding accuracy of the IHDST is analyzed in detail. Accordingly, a working parameter optimizing strategy is developed for high centroiding accuracy and improved dynamic performance. Finally, the laboratory tests and the night sky experiment support the conclusions.
© 2017 Optical Society of America
1. Introduction
A star tracker is a high-precision instrument, provides three-axis attitude, and is increasingly applied in spacecraft [1]. However, the start tracker is limited by its dynamic performance of generally less than 3°/s [2]. Under dynamic conditions, the star spots are smeared out over several pixels while the spacecraft is slewing and the intensity of the pixel decreases. Thus, extracting the star from the low signal-to-noise ratio (SNR) image is difficult, and the star locating accuracy is reduced. Consequently, the attitude accuracy and valid update rate are decreased [3–5].
The solution to this problem restricts the smearing effect by shortening the exposure time, and the drop of the intensity is compensated by improving the star detectability. StarVison company’s SG100 star tracker uses an image intensifier in an image-intensified charge-coupled device (ICCD) based imaging system and achieves a dynamic performance of up to 20°/s [6]. The image intensifier significantly improves the star detection sensitivity during short exposure time by multiplying the input starlight signal. Thus, the dynamic performance is significantly improved. We call the star tracker that uses an image intensifier to achieve high dynamic performance as the intensified high dynamic star tracker (IHDST) and investigate its imaging character and star locating accuracy.
The imaging process of an electro-optical imaging system can be described as quantum (photon or electron) transfer process and spatial spreading process [7]. The quantum transfer process is a stochastic process caused by the randomness of the input quantum and multiplication. The output of such system is a random variable with mean and variance of signal and noise, respectively. The imaging process of a traditional star tracker is relatively simple. The starlight passes through the lens first and then photoelectrons are generated in the image sensor. The number of the photoelectrons generated in each pixel is still subject to the Poisson distribution. Therefore, the output signal of the system is the average number of the input photons, and the noise is considered the shot noise. The spatial spreading process can be described by a Gaussian function, and the spreading area is usually in the range of 3 × 3 pixels to 5 × 5 pixels [1,8].
Given that the imaging system of IHDST uses an image intensifier, the imaging process is complex. The input photon goes through many stages, such as optical lens focusing, photoelectric conversion, electron multiplication, fluorescence excitation, fiber coupling transmission, and conversion to digital star image [9]. The quantum transfer characteristic of IHDST is more complicated than that of a traditional star tracker and should be modeled as a compound random variable. The spatial dispersion of IHDST is also different from the Gaussian function, and the spatial spreading area increases to 7 × 7 pixels. Obviously, the imaging model of a traditional star tracker cannot accurately describe the imaging character of IHDST. Therefore, an accurate model of the IHDST imaging process must be established.
In [9], Katake first established a star imaging model for the ICCD used in the star tracker. However, the gain at each stage of the imaging process is considered a deterministic gain rather than a stochastic gain. Thus, the estimation for the gain is inaccurate. In [7], [10], and [11], researchers used the cascade model to study the quantum transfer character of each stage in the imaging process, and they derived theoretical models for evaluating the imaging performance of a system. However, these models focus on the variance of the output signal and fail to provide the complete probability distribution of the output signal. Therefore, the output signal should be assumed to be subject to a certain distribution when simulating the star imaging process. Hence, the abovementioned models are still unsatisfied. For this reason, we establish a quantum transfer model of IHDST based on stochastic process theory and derive the probability distribution function (PDF) of the output quantum number of the imaging system. We introduce two-dimensional Lorentz functions to describe the spatial spreading process, and the spatial spreading character of the IHDST is modeled as the convolution of several point spread functions (PSFs). Considering the interaction of the spatial spreading character and the quantum transfer character, a complete star imaging model of IHDST is provided.
The three-axis attitude of a star tracker is obtained by locating the positions of star spots in image plane; therefore, the accuracy of the star spot locating directly determines the attitude accuracy of the star tracker [3]. On the basis of the imaging model of IHDST and the Monte Carlo method, the influence of the star tracker working parameters, such as exposure time and gain control voltage, on the star locating accuracy is analyzed in this study. Then we develop a strategy for setting the working parameters of an IHDST to maintain its high-precision star locating accuracy under high dynamic condition. Finally, the laboratory test verifies the analysis of the star locating accuracy, and the night sky experiment verifies the correctness of the working parameter setting strategy of IHDST.
2. Theory and method
The signal and noise transfer model of a traditional star tracker is relatively simple. Therefore, the imaging model under dynamic conditions mainly focuses on the spatial spreading of star energy (motion blur). However, the quantum transfer and spatial spreading characteristics of an IHDST are different from those of the traditional star tracker. Therefore, a new imaging model for the IHDST must be developed. In this section, the principle of IHDST is first described in detail. Then, the quantum transfer and spatial spreading models are established. Finally, the strategy of setting working parameters is obtained by analyzing the star locating accuracy of IHDST.
2.1 Principle of IHDST
The principle of IHDST for attitude measurement is the same as that of a traditional star tracker. When a star image is obtained, the star position can be calculated by the star locating algorithm. Then, each star observed in the image is compared against a firmware star catalog [1]. Each star is identified with the help of the star recognition algorithm. Therefore, we obtain the equation: Wi = AVi, where Wi is the observation star vector, Vi is the reference star vector, and A represents the attitude matrix. We solve the equation using the attitude determination algorithm to obtain the three-axis attitude of star tracker.
The imaging system of IHDT is composed of the optical lens, the image intensifier, the fiber optic taper (FOT), and the image sensor, as shown in Fig. 1. The image intensifier comprises the photocathode, the microchannel plane (MCP), and the phosphor screen.
The process of the IHDST imaging is as follows. First, the starlight passes through the optical lens and forms an image at the surface of the photocathode. These photons are converted to photoelectrons in the photocathode owing to the photoelectric effect. These photoelectrons are accelerated and run into the MCP wherein they are multiplied. When multiplied electrons are accelerated by a high-voltage electric field and strike on the phosphor screen, the phosphor screen releases photons and forms a photon image. The photon image is transmitted to the image sensor by the FOT. Finally, the photon image is converted to a digital image by the image sensor.
The high sensitivity of IHDST mainly depends on the electron multiplication of MCP [12]. MCP is a slightly conductive glass substrate with millions of parallel traversing microchannels containing a secondary electron emitter on their inner walls. Photoelectrons (primary electrons) emitted by the photocathode under high-voltage electric field enter the MCP, wherein these photoelectrons strike the tube wall and excite the secondary electrons. This process is called the first multiplication. The secondary electrons repeat this process and excite additional secondary electrons, and this process is called the secondary multiplication. After a multi-stage multiplication, the factor of total multiplication in the MCP achieves 103 to 105, as shown in Fig. 2.
In the figure, λ1 is the primary multiplication factor, δi (i = 1, 2, ⋯, M) is the secondary multiplication factor, and M is the number of stages of the secondary multiplication. The MCP multiplication can be regarded as a series of cascaded stages. The accelerating voltage of the primary stage is larger than that of the secondary stage; thus, the primary multiplication factor is higher than the secondary one, λ1 > δi. All the accelerating voltages of the secondary stages are assumed to be equal. The secondary multiplying factors are also assumed equal, that is, δi = δ. In practice, the high-voltage U (600 V to 900 V) across the MCP channel is controlled by another low-voltage Uc (0 V to 5 V), which is called gain control voltage. Thus, δ is a function of Uc. The mean gain of the MCP is described as [12]
In [12], Eberhardt measured a typical MCP and provided a set of parameters with λ1 = 2.02, δ = 1.39, and M = 15.2; thus, gMCP = 300.
2.2 Statistic of compound stochastic process
The IHDST imaging process includes multiple stages of photoelectric conversions and electron multiplications. The imaging system is modeled as a series of cascaded single stage [7,9–11]. If the input is a single quantum (a photon or an electron), then the number of output quantum is random owing to the quantum behavior of photons and electrons in each stage. Therefore, the gain of each stage is a discrete random variable g. The mean and standard deviation of g are denoted as μg and σg, respectively. Considering that the process of each IHDST stage is a discrete stochastic process, the total imaging process is a compound stochastic process. In studying the properties of compound stochastic process, we define the probability generating function (PGF) of random variable g as
where p(x) is a PDF of g, which is uniquely determined by its PGF and vice versa [13]. Given a stage with the gain of g, the k-th quantum (k = 1, 2, ⋯, N) is multiplied by g(k), where N is the number of input quantum and is a random variable. If we denote S as the output of the stage, then S is the sum of N random variables, S = g(1) + ··· + g(k), which is called compound distribution of g under N. Considering that S ≠ Ng, the PGF of S is expressed as [13]The PDF of S is obtained as
N-order derivation consumes much computing; thus, we use the Panjer recursion algorithm [14] as follows:
When N is subject to a Poisson distribution with the parameter λ, then a = 0, b = λ. When N presents a negative binomial distribution with the parameters r and p, then a = 1 − p, b = (1 − r)(1 − p).
If N is subject to a Poisson distribution with the parameter λ, and g is subject to a Bernoulli distribution with the parameter η, then S follows the Poisson distribution with the parameter λη [15]. This inference can simplify the modeling process of IHDST.
The mean and variance of S are expressed as
Equation (7) show that, if the gain g is constant (σg = 0), then σS = σNμg, which means that the output SNR is equal to the input SNR. However, if the gain g is a random variable, then the multiplication process introduces additional noise called the gain fluctuation. This fluctuation results in that the image noise of the IHDST is much larger than that of the traditional star tracker.
If the random variable N is regarded as the output of the previous stage with a single quantum input, which means that the gain of this stage is N, and the gain of the next stage is g, then S can be regarded as a two-stage cascaded multiplication system with its PGF represented as Eq. (3). Supposing that the gain of the k-th stage is gk (k = 1, 2, ⋯, n), then the PGF of the n series of cascaded system is expressed as
The mean of the cascaded system is as follows:
where μk is the mean of the k-th gain. The variance of the cascade system is iterated by Eq. (7), and its PDF can be obtained by Eq. (5). Given that the PDF of the cascaded system is known, the mean, variance, and other parameters, such as the noise factor, detective quantum efficiency, and SNR, can be easily derived. Hence, PDF is a comprehensive description of a cascaded system.2.3 Star spot quantum transfer model
The star energy of the IHDST input can be expressed as the number of incident photons, and the photons transfer process of the IHDST star spot is regarded as the quantum transfer process. The IHDST imaging process is decomposed into a series of imaging processes. Each of the IHDST imaging processes can be considered an independent stochastic process, such as the Bernoulli, Poisson, Polya, and Geometric processes [15]. Their PDF, PGF, mean, and variance values are shown in Table 1.
Table 1. Typical stochastic processes
The output of a single quantum through the stage is either 0 or 1 in a non-multiplication process. In [16], Westmore et al. denoted this process as a binary selection process and derived the signal and noise power spectrum. This process is in fact a Bernoulli process. The behavior of a single quantum that passes through the imaging stage and outputting 1 can be considered a Bernoulli trial with the probability of success η (η < 1). In [7], Yang et al. modeled this process as a Binomial process, which consists of a fixed number n of statistically independent Bernoulli trials. However, the number of input quantum n is commonly stochastic, and satisfying the condition that n is constant is difficult. Hence, the Bernoulli process is an accurate model.
The output of a single quantum through the stage can be 0, 1, 2, ⋯, or ∞ in a multiplication process. The behavior of electrons in the MCP is a typical multiplication process. Given the different manufacturing processes of the MCP, the single-stage multiplication process can be a Poisson, Polya, or geometric process. The Polya distribution is also called the negative binomial distribution, which is a distribution between the Poisson and geometric distributions. The Polya distribution becomes the geometric distribution when r = 1, while the Polya distribution becomes the Poisson distribution when r → ∞. Therefore, the multiplication process of a single stage in MCP is assumed as a Polya process [13,17]. Figure 3 shows the PDF of the geometric gain gGeo, Polya gain gPol (r = 2), and Poisson gain gPoi. The said figure shows that the mean of the three gains are the same (μ = 9), but their variance (gain fluctuation) is different. Geometric gain fluctuation is the largest one, Polya is the second, and Poisson is the smallest. Thus, we can reasonably infer that the parameter r describes the gain fluctuation.
The number of stellar photons that arrive at the surface of the optic lens during the exposure time T is a random variable Pin(λinT), which is described by a Poisson distribution [3]. λin indicates the rate at which the optical lens with aperture D collects incident photons and is expressed as [1]
where Mv is the star apparent magnitude, E0 is the irradiance of a star of magnitude Mv = 0, E0 = 2.96 × 10‒14W/mm2 [1], and ελ is the energy of a photon with wavelength λ.
The transmission process of incident photons is modeled as a Bernoulli process with τ1 as the lens transmittance. The process of photons converted into photoelectrons in the photocathode is modeled as a Bernoulli process with the parameter η1, which is the quantum efficiency of the photocathode. Section 2.2 implies that the compound stochastic process of the three cascaded stages is equivalent to a Poisson process with the parameter λ0 (λ0 = λinτ1η1T), and the PDF of the latter process is expressed as
The primary multiplication of the MCP is modeled as a Poisson gain process with the parameter λ1, which is the mean gain of the primary multiplication [7]. All the secondary multiplications of MCP can be regarded as a compound Polya process denoted as CMCP2(δ, r, M), where δ is the mean gain of each secondary multiplication, r is the parameter that describes the gain fluctuation, and M is the stage number. Assuming that the Polya gains are independent and identically distributed, the PGF of the compound Polya gain is derived as
where GM−1(s) is the PGF of the M−1 series cascaded Polya gains, and G(s) is the PGF of a single Polya gain process. The PGF of the total compound gains of MCP is derived asThe multiplied electron is then converted back into a photon at the phosphor screen. This process is modeled as a Poisson process with the parameter λ2, which is the quantum efficiency of the phosphor screen. The process of a photon that passes through the FOT is a Bernoulli process with the parameter τ2, which is the coupling efficiency. Finally, the photons are converted into electrons in the image sensor pixels. This process is a Bernoulli process with the parameter η2, which is the quantum efficiency of the image sensor. Similarly, the cascaded stochastic process of the three stages is equivalent to a Poisson process with the parameter λ2τ2η2, and the PGF of the latter process is expressed as
Thus, photons from a star is multiplied by the IHDST and converted into electrons in the image sensor. The electron number is modeled as a compound stochastic process with its PGF expressed as
Its PDF pST(x) is obtained by Eq. (5). Finally, the pixel electrons are converted into a digital image by ADC, and the conversion gain K is a constant. The PDF of the digitalized image gray value is derived as
where qk is the minimum electrons for digitization of k, qk = k/K. The mean gain of the IHDST imaging system isIf the input of an IHDST is the starlight, then the output of the IHDST is the digital image signal expressed as
Thus far, each of the IHDST imaging stages is modeled as a certain stochastic process, and the quantum transfer model of the IHDST is a compound stochastic process. Given the system parameters and input photon number, the probability distribution of the IHDST output can be accurately simulated.
Most of the parameters of the quantum transfer model can be found in the datasheet provided by the manufactures. However, a few detailed parameters of the image intensifier are unknown, including the primary multiplication coefficient λ1, secondary multiplication coefficient δ, and gain fluctuation factor r. The stage number of the MCP secondary multiplication M is mainly determined by the length and diameter of the microchannel and its materials. Once the MCP is made, M is a constant. We chose the typical value of M = 15 in our experiment as the manufacture provided. Given that the MCP is encapsulated in an image intensifier, it cannot be easily measured by the user [7]. Hence, we estimate the MCP parameters using our quantum transfer model. In [12], Eberhardt provided reasonable estimations of the MCP parameters, except the parameter r. Therefore, a new method is developed to obtain accurate parameters.
If the star magnitude and exposure time are given (e.g., Mv = 6.5, T = 2 ms), then the mean number of the input photons is determined as 150 by Eq. (10). These photons are multiplied and converted into a digital star image as the IHDST output. We set Uc = 3.8 V, and take 1,000 star images by the IHDST. The star energy is expressed as the pixel intensity denoted as Itot,k (k = 1, 2, ⋯, 1000). Given the gain fluctuations, Itot,k varies and shows a certain distribution. The probability distribution of the star energy, which is denoted as pex(x), is obtained by normalizing the histogram of Itot,k (Fig. 4). The mean of Itot,k is then calculated as 276.4. The MCP gain is obtained using Eqs. (1), (17), and (18), and gMCP = 162.1. We can derive the PDF of the output star signal theoretically, which is denoted as pth(x; λ1, δ, r), on the basis of the quantum transfer model and Eqs. (5), (15), and (16). Given that M = 15 and gMCP = 162.1, then δ = (gMCP/λ1)1/15. Thus, the parameters λ1 and r must be estimated as follows:
The experimental PDF pex(x) is expressed as the histogram in Figs. 4(a) and 4(b). λ1 takes different values when r = 2, and the theoretical PDFs are shown as the colored lines in Fig. 4(a). Similarly, r takes different values when λ1 = 4.04, and the theoretical PDFs are shown as the colored lines in Fig. 4(b). The said figure shows that the theoretical value best fits the experimental data when λ1 = 4.04 and r = 2.
2.4 Star spot spatial spreading model
A signal obtains a certain degree of a spatial spreading process when it passes through an optical system. This characteristic is generally described by PSF or modulation transfer function [7,9]. An IHDST imaging process includes the following spreading processes: optical lens spreading, front and back proximity focusing bi-planar electron lens spreading, and FOT spreading caused by the coupling of the intensifier and image sensor. First, the PSF of the optical lens can be approximated by a Gaussian function as follows [1‒3,8]:
where ρOL is the Gaussian radius that indicates the spreading scale. When the photoelectrons emitted from the photocathode are subject to the Maxwellian velocity distribution, the PSF of the front bi-planar electron lens is also a Gaussian function denoted as fF(x, y; ρF) with a radius ρF [18]. Given that the convolution of two Gaussian functions is still a Gaussian function, the PSF of the optical lens and front bi-planar electron lens is equivalent to the following:where . The PSF of the back bi-planar electron lens is different from the Gaussian function because the velocity of the multiplied electrons emitted from the MCP has no regular pattern. This PSF is commonly determined experimentally and is modeled as a Lorentz function as follows:where ρB is the Lorentz radius, and ρB is a function of the MCP gain control voltage Uc. Coupled with the image sensor, the PSF of the FOT can also be expressed as a Lorentz function, fFOT(x, y; ρFOT) [19].The spatial spreading model of a traditional star tracker is simply modeled as the convolution of the optical lens spreading function fOL(x, y; ρOL) and trajectory function fin(x, y), which describes the motion of the star on the image plane. Similarly, the star spreading function of IHDST is modeled as
where ⊗ denotes the convolution operation, λ0T is the mean number of photons collected by the lens in the exposure time, and gST is the mean gain of IHDST. The grayscale of the star image at pixel location (i, j) isThus far, the spatial spreading model of the IHDST is established, which is the theoretical value of the star image without noise.
2.5 IHDST imaging model
For a traditional star tracker, the stellar photons are converted into electrons in the image sensor pixels. The photoelectron number in every pixel is independent of each other and is subject to the Poisson distribution. The quantum transfer model of a traditional star tracker is simple such that its spatial spreading model is mainly considered. However, the signal and noise distribution of each pixel for an IHDST are affected by the interaction of the quantum transfer process and spatial spreading. Thus, we should establish the IHDST imaging model considering both the quantum transfer process and spatial spreading.
The star spot image at the MCP front surface is expressed as a 2D Gaussian function f1(x, y; ρ1). If this star spot covers N microchannel tubes, then A = ΣNAk, where A is the spreading area of this star spot and Ak is the k-th microchannel area. The k-th microchannel incident photoelectron rate is as follows:
The total number of incident photoelectrons of the MCP is subject to Poisson distribution, and the mean of this number is λ0T. Thus, the number of input photoelectrons of the k-th microchannel is also subject to the Poisson distribution according to the Poisson distribution property [15] as follows:
where fPoisson(x) is a random number generation function, and the number it generates is Poisson distributed. Each of these nk photoelectrons is multiplied by the k-th microchannel, and the PDF of the MCP gain pMCP(n) can be obtained from Eqs. (13) and (5). The number of multiplied electrons for the i-th input photoelectron is obtained using the Alias Method [20] as follows:where fAlias(p) is a random number generation function, and the probability of the number n generated by the function is p(n). After all of these nk photoelectrons are multiplied by this microchannel, the total number of output electrons is expressed asIf this simulation is conducted for all the N microchannel tubes within the spreading area A, and the spatial arrangement of the microchannel tubes remains unchanged, then the multiplied electronic image IMCP at the MCP back surface can be obtained. The said scenario implies that IMCP is an arrangement of Nk, IMCP = {Nk}, k = 1, 2, ⋯, N. The image IMCP is blurred by the back bi-planar electron lens, amplified by the phosphor screen, spread and attenuated by the FOT, and finally digitalized by the image sensor. The final star image is modeled as
This model describes the IHDST imaging process and particularly describes the noise caused by gain fluctuation.
2.6 Analysis of the star spot locating error
Given that the attitude calculation of star tracker is achieved by locating the position of the star spot on the image plane, the accuracy of the star position is directly related to its attitude accuracy. The true position (xt, yt) can be estimated by different algorithms. However, not all algorithms, such as Gaussian fitting algorithm, are suitable for the smeared star spots under dynamic conditions. Thus we use the centroid algorithm to estimate the star position. Firstly, we explain why we choose the Monte Caro method to evaluate the star centroiding error. Then the centroiding error for IHDST is analyzed under static and dynamic conditions, respectively.
A. Method to evaluate centroiding error
The centroid algorithm [1] is expressed as
where n is the pixel number with a grayscale larger than the threshold, Ik and (xk, yk) are the gray value and coordinates of the k-th pixel, respectively. The centroid (xe, ye) is the estimated value of (xt, yt). However, an error between the estimated and true values exists because of the pixel noises [8]. The centroid error in x-coordinate is derived aswhere Itot is the summation of the pixel gray values, Itot = ΣnIk, ρij is the noise correlation coefficient of the i-th pixel and j-th pixel, and σIk is the k-th pixel noise. The expression of σy is similar to that of σx such that it is omitted here. The centroid location error can be decreased by increasing the total grayscale Itot, and lowering the pixel noise σIk and the noise correlation coefficient ρij as exhibited in Eq. (31).An analytical expression for a traditional star tracker can be accurately derived by Eq. (31) [3]. However, deriving an analytical expression to analyze the centroid error of IHDST is difficult. First, the pixel noise σIk is no longer subject to the Poisson distribution because of the gain fluctuation. Second, the noise of adjacent pixels is dependent because of the spatial spreading (ρij ≠ 0), and the exact value of ρij is difficult to determine. Third, the spatial spreading model is also a complicated convolutional expression, so the pixel signal is difficult to determine by an analytical expression. Thus we use the Monte Carlo method to quantitatively study the centroid location accuracy of IHDST and solve this problem, see Fig. 5. This method is based on the proposed imaging model.
Fig. 5 IHDST star image and its locating accuracy. (xe,i, ye,i) is the centroid of the i-th star image. σx and σy are the star locating accuracy in the x- and y-directions, respectively.
We let the star spot be imaged by N times at the same position on the image plane, and their centroids are obtained as (xe,i, ye,i), i = 1, 2, ⋯, N. If N is sufficiently large, then the mean (‾xe,‾ye) of the centroids approximates the true position of the star spot. The locating accuracy of the star is expressed as the standard deviation of the centroid [21] as follows:
The total error of the star centroid is as follows:
B. Centroiding error under static conditions
The main factors that affect the star imaging are the incident star magnitude Mv, star tracker angular velocity ω, exposure time T, and gain control voltage Uc. The detection threshold of the star magnitude for a typical star tracker is commonly set to Mv = 6 [2]. Thus, the star number within the field of view (FOV) meets the requirements of the all-sky star identification and storage of the star catalog. Therefore, the following analysis of the centroid error is based on Mv = 6. First, the relationship among the star locating error, exposure time T, and gain control voltage Uc is analyzed. T is set as 2, 4, 8, and 16 ms, and Uc is in the range of 3.0 V to 4.0 V. Approximately 1,000 star spots are generated according to the IHDST imaging model under each condition, and their centroids are then calculated. The centroiding errors under different conditions are shown in Fig. 6.
The gain remains constant (i.e., Uc = 3.5 V), as shown by arrow b in Fig. 6. The total grayscale of the star image Itot increases when the exposure time increases. The centroid error gradually decreases with the increase in exposure time. The reason is that the influence of Itot on the centroiding error exceeds the influence of the pixel noise σIk.
The IHDST gain is small and the star spot is excessively dark to be accurately located if the exposure time remains constant when Uc is small (area a of Fig. 6). The IHDST gain increases to several hundred times when Uc is large (area c of Fig. 6). Certain pixels of the star spot in the image then saturate such that the centroiding error increases. The centroiding error tends to decrease slightly with the increase in Uc outside regions a and c.
Extending the exposure time is commonly more effective than increasing the Uc to improve the accuracy of the star centroid. We set up three cases to further validate this conclusion. Each case has different exposure times and gain control voltages. Case 1 represents a shorter T and a higher Uc. Case 2 represents a moderate T and Uc. Case 3 represents a longer T and a lower Uc. The total grayscale Itot of the star image remains the same in all cases. A total of 1,000 star points are generated for each case using the imaging model of IHDST. The centroid of each star is then calculated, and the centroiding error is shown in Table 2.
Table 2. Centroiding errors of the three static cases
Table 2 shows that, if the grayscale of the output star remains constant, then the centroiding error of Case 3 is the smallest. Therefore, we should extend the exposure time first, and then set an appropriate gain such that the star image brightness is close to saturation to improve the accuracy of centroid positioning.
C. Centroiding error under dynamic conditions
The star spot is smeared under the dynamic condition, and its centroiding error is different from that of the static condition. Hence, the centroiding error is studied separately under the dynamic condition. The simulation conditions are set as follows: the stellar magnitude is 6, the angular velocity is 20°/s, the rotation axis of the IHDST is parallel to the y-axis of the image plane, and the star spot moves along the x-direction of the image plane.
If the gain control voltage Uc is fixed at 3.8 V when the exposure times T are 1, 2, 3, 4, 6, 8, and 16 ms, then the centroiding error varies with the exposure time (Fig. 7).
The star spot is blurred in the x-direction, and the smear length is proportional to the exposure time. When the exposure time is short, the total grayscale Itot of the star spot is small, and the centroiding error σx derived by Eq. (31) is large. The demerit of the smear effect overcomes the merit of the increasing Itot, and the centroiding error σx is large when the exposure time is long. As shown in Fig. 7, the centroiding error σx tends to decrease first and then increase with the increase in exposure time. As the star spot is not smeared in the y-direction, the centroiding error σy decreases when the exposure time increases. This trend is similar to that in the static condition. The total star locating error σc is similar to σx. σc decreases first and then increases as the exposure time increases.
If the exposure time is fixed at 3.4 ms and the gain control voltage Uc is in the range of 3.1 V to 4.0 V, then the centroiding error varies with Uc (Fig. 8). Figure 8 shows that the intensity of the star spot is weak, and the centroiding error is large when Uc is small. The intensity of the star spot gradually increases, and the centroiding error decreases when Uc increases. Therefore, as the gain control voltage Uc increases under the dynamic condition, the centroiding error σx in the x-direction gradually decreases whereas σy is nearly unchanged. The total star locating error is similar to that of σx. Specifically, σc decreases with the increase in Uc.
The analysis of the star locating accuracy under static and dynamic conditions shows that the optimal star locating accuracy can be acquired by setting a reasonable exposure time and gain control voltage.
2.7 IHDST working parameters setup
The previous subsection shows that the exposure time T is the main parameter that affects the star locating accuracy. Thus, the exposure time should be the first parameter to optimize. The highest star locating accuracy is then achieved by setting a proper gain gMCP, which is controlled by the gain control voltage Uc.
We suppose that the star tracker rotates at a fixed angular velocity ω; if the rotation axis is perpendicular to the boresight of the star tracker, then the smear effect is at its most serious level. The star motion in the image plane can be approximated as a uniform linear motion, and the trajectory is approximated to a straight line under the aforementioned condition [3]. The intensity profile of the star spot when the exposure time gradually increases from zero is shown in Fig. 9.
Figure 9 shows that the star spot moves at the velocity v along the x-axis, and the vertical axis indicates the intensity of the star spot. The exposure starts at time t0, and the star spot spreads to a region with center and radius of x0 and r, respectively. The star spot moves to x1 at time t1. The star energy is accumulated from time t0 to t1 at position p in the image plane such that the intensity at p gradually increases. This process is illustrated as I1 in Fig. 9. However, the spreading area cannot cover position p if the star spot moves beyond x1; thus, the energy intensity at p does not increase during the period from t1 to the end of the exposure t2. This process is illustrated as I2 and indicates that the long exposure time makes the star spot smeared, but the intensity remains unchanged.
The analysis of the centroiding error in the previous section states that, when the smear length is extended from zero to L1(L1 = 2r), the intensity of the star spot gradually increases. Furthermore, the centroiding error gradually decreases and reaches its minimum. However, when the smear length is longer than L1, the star spot smears out without the increase in intensity. The centroiding error also begins to increase. Therefore, the dynamic exposure time is determined as
where v represents the velocity of the star spot in the image plane in units of pixel/s. Npix is the square root of the number of active pixels. r is the IHDST spreading radius, 2r = 7 pixels. When the angular velocity is small, the exposure time obtained by Eq. (34) is larger than the upper limit of the exposure time Tup, which is the reciprocal of the star sensor data update rate (f Hz). When the data update rate of IHDST is 50 Hz, Tup = 20 ms. Therefore, the relationship between the exposure time and angular velocity is shown in Fig. 10.We suppose that Uc is set at 3.6 V and the stellar magnitude Mv is 6; thus, the exposure time is set according to the angular velocity ω as shown in Fig. 10. The intensity of the star image along the moving direction is shown in Fig. 11(a). The star spot is nearly saturated under the static conditions, while the intensity of the star spot is gradually decreased under the dynamic conditions. The brightest pixels of the star spot at different angular velocities are shown in Fig. 11(b).
Figure 11 shows that, when the angular velocity increases, the brightness of the star image decreases. Nevertheless, the intensity of the star image and centroiding accuracy increase if we increase Uc. Therefore, Uc should be increased to compensate the weakening of the intensity caused by rotation such that the centroiding accuracy can be improved. When ω = 10°/s, Fig. 10 shows that the exposure time is set to 7 ms, and Fig. 11(b) shows that the brightest pixel of the star spot decreases to 44 DN. However, if Uc increases to 3.96 V, then the star intensity returns to near saturation, and the highest star locating accuracy is acquired. The settings of the gain control voltage are shown in Fig. 12.
Figure 12 shows that the optimizations for each star magnitude are different. When the angular velocity ω is less than 10°/s, Uc is set as the blue curve such that the star intensity of magnitude 6 always remains nearly saturated. When ω is larger than 10°/s and smaller than 25°/s, Uc is set as the red curve such that the star on Mv = 5 is always maintained near saturation. When 25°/s < ω < 40°/s, Uc is set as the yellow curve, and the stars brighter than magnitude 4 can still be detected.
If the design goal for a star tracker requires the dynamic performance of 40°/s, then the optimization of the working parameters is focused on the stars on Mv = 4. If the upper limit of the slew rate is 10°/s but the attitude accuracy must be high, then the target star for designing a star tracker should be the stars on Mv = 6. Uc does not have to be tuned along the red line to satisfy the different requirements in different cases. The region between the blue and yellow line is also an alternative.
3. Experiments and results
3.1 Experiment setup
The hardware configuration of the IHDT imaging system is shown in Table 3, which lists the main parameters of the optical system, image intensifier, and image sensor.
Table 3. IHDST hardware configurations
The Laboratory setup is shown in Fig. 13. The IHDST is mounted on a turntable, and the boresight of the IHDST is aligned with the star simulator. The star spot is then imaged at the center of the image plane to avoid lens distortion. The input starlight energy is controlled by changing the star magnitude of the star simulator and exposure time of the IHDST. Given the number of input photons and gain of the image intensifier, 1,000 star images are collected. Each image is then sampled under a set of parameters, such as the exposure time T, gain control voltage Uc, and magnitude Mv of the star simulator.
3.2 Validation of the IHDST imaging process
First, the quantum transfer model of the IHDST is validated. Four groups of the experimental data are obtained under the following conditions: Mv = 7, T = 4 ms. Uc is set to 3.5, 3.6, 3.7, and 3.8 V. The total grayscale of each star image is counted, and the histogram is obtained for each group of the experimental data. Thus, its PDF is shown as the stair curve in Fig. 14. The theoretical PDF of the IHDST output is obtained from the quantum transfer model, as shown in the solid curve in Fig. 14. The said figure shows that the experimental values agree well with the theoretical values; this agreement indicates that our quantum transfer model accurately describes the quantum transfer characteristics.
The spatial spreading model of IHDST is then verified as shown in Fig. 15. Three groups of the experimental data are obtained under the conditions Mv = 6, T = 2 ms, and Uc is set to 3.8, 3.9, and 4.0 V. Each group contains 1,000 star images, and the mean of these images approximate to a noise-free image, which represents the spatial spreading of the star spot. The said figure shows that the three star spots in the first row are the experimental data. The star spots of the second row are generated using the spatial spreading model in this study. The star spots of the third row are generated by the Gaussian PSF model. Calculating the root-mean-square error (RMSE) between the simulated and experimental star spots (Fig. 15) shows that the star spots generated by our model possess a smaller RMSE than the ones generated by the Gaussian model. For example, the RMSE calculated using our model is 1.52 when Uc = 3.8 V, which is less than the RMSE (1.78) of the Gaussian PSF model. Thus, our model describes the spatial spreading accurately.
Finally, the imaging model of IHDST is verified. The IHDST character is evident under the low input light intensity, short exposure time, and high gain conditions. We select Mv = 6, T = 2 ms, and Uc = 4.0 V as the simulation and experimental conditions. The star spots imaged every time are significantly different from one another because of the IHDST gain fluctuation. Thus, a single star spot is insufficient for verification. Hence, we select 25 experimental star spots and 25 simulated star spots, which are shown in Figs. 16(a) and 16(b), respectively. Moreover, we simulate another 25 star spots based on the star imaging model in [9] for comparison, see Fig. 16(c).
Fig. 16 Validation of the IHDST imaging model. (a) Experimental star spots; (b) Simulated star spots based on our model; (c) Simulated star spots based on the model in [9].
The brightness of stars in Fig. 16(a) varies due to the gain fluctuation, and Fig. 16(b) well simulates this noise. The stars in Fig. 1(b) are similar to those in Fig. 1(a) which means that our model is a well description to the real situation. By contrast, the stars in Fig. 1 (c) seem so “ideal” and differ far from the experimental star spots. Thus our model apparently reflects the gain fluctuation noise and is a better model.
Hence, the imaging model of IHDST is fully validated and properly describes the imaging character of IHDST.
3.3 Star centroiding accuracy verification
The effect of exposure time and gain control voltage on the star locating accuracy is studied on the basis of the IHDST imaging model and Monte Carlo method. Experiments are then conducted to verify the conclusions about the centroiding accuracy.
The experimental conditions are the same as those in Section 2.6. Mv = 6, T is 2, 4, 8, and 16 ms, and Uc is in the range of 3.0 V to 4.0 V. The centroids of the experimental stars in each group are calculated by Eq. (30), and then the centroiding error under different conditions is obtained by Eq. (32) (Fig. 17).
Taking the case of T = 4 ms as an example, the star spot is excessively dim to extract from the background image when Uc < 3.3 V. The central pixels of the star spot are saturated, and the centroiding error increases when Uc > 3.9 V. However, the centroiding error of the simulated star spot for 3.3 V < Uc < 3.9 V is nearly the same as that calculated by the experimental data.
The conditions for the second experiment are also the same as those in Table 2. The centroids of the star in each case is obtained as (xi, yi)exp, i = 1, 2, ⋯, 1000. The centroiding error of the different conditions can then be obtained from Eq. (32) (Table 4). Table 4 lists the theoretical and experimental results of the centroiding error in different cases, which agree well with each other.
Table 4. Simulation and experiment results of the centroiding error
The two experiments above show that the centroiding accuracy of the IHDST can be accurately estimated by the Monte Carlo method using the star imaging model established in this study.
The star spot moves across the image plane under the dynamic condition. Ensuring that the star spot is captured at the same position every time is difficult. Therefore, the true position of the star is unknown, and Eq. (32) fails to estimate the centroiding error. However, the dynamic centroiding error can be verified indirectly through the night sky experiment. When the star locating error increases, the star spot fails to locate, the star cannot be identified, and the attitude of IHDST is lost. When the star locating error is small, the attitude output is stable. Therefore, qualitatively verifying the accuracy of the star locating accuracy is possible by judging whether the attitude outputs are stable. This experiment is demonstrated in the next subsection to save space.
4.4 Validation of parameter optimization
We conducted a night sky experiment at the Xinglong Station (National Astronomical Observatories, China) to verify the working parameters of IHDST (Fig. 18). The IHDST was mounted on a portable high-precision turntable and aligned at the zenith during the night sky experiment. The turntable speed was set to 8°/s, 12°/s, 16°/s and 32°/s. The exposure times were determined to be 8, 6, 4, and 2 ms in accordance with the parameter setting method described previously.
Table 5 shows the states of the IHDST tracking performance under different combinations of angular velocity ω, exposure time T, and gain control voltage Uc.
Table 5. Tracking state under different conditions
Table 5 denotes that “Track” indicates the stable tracking state, whereas “Fail” indicates a failure of tracking and that the three-axis attitude cannot output stably. Table 5 also indicates that, when Uc is low, the star spot is excessively dark to extract sufficient stars for star identification, the star locating error is large, and stabilizing the output is difficult. The star spot is saturated when Uc is excessively high, while the stray light is amplified. This scenario introduces additional noise and can result in a misidentification of the star and unstable attitude output. If the parameter is properly configured according to the parameter setting method, then IHDST can achieve stable tracking. Therefore, the parameter setting method is proved to be extremely useful, and the analysis of the centroiding accuracy is reasonable.
4. Conclusions
The principle of IHDST is first introduced in this study. Given the multiplication of the incident starlight, high sensitivity of the star detection is achieved under short exposure times such that an extremely high dynamic performance is achieved. A complete model is then established for the imaging process of the IHDST. The model describes the quantum transfer process and spatial spreading of the imaging system. The quantum transfer process of the IHDST can be expressed as a compound stochastic process. The spatial spreading of the system is modeled as the convolution of every PSF. The centroiding accuracy of the IHDST under static and dynamic conditions is analyzed in detail on the basis of this model. The exposure time and gain of the IHDST affect the centroiding accuracy, and the exposure time is the primary factor. Thus, a working parameter optimizing strategy is developed for high centroiding accuracy and improved dynamic performance. Given that the angular velocity is known, the exposure time is first set to avoid a smear. The intensity loss of the star image because of the smearing is compensated by increasing the gain control voltage. Finally, the model correctness and analysis of the centroiding accuracy are verified by a laboratory test experiment. The correctness of the IHDST working parameters is verified by a night sky experiment.
Funding
National Natural Science Foundation of China (NSFC) (61222304); Specialized Research Fund for the Doctoral Program of Higher Education of China (20121102110032).
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