1. Introduction

A star tracker is a high-accuracy attitude estimation instrument for spacecraft. It consists of an imaging unit and a processing unit. During the imaging stage, stellar light passes through the optical lens of a star tracker and forms a star image. In the following stage, the processing unit performs star spot location estimation, star identification or tracking, and attitude determination. Finally, a star tracker outputs high-precision three-axis attitude information [1]. With the development of space technology, the dynamic performance of a star tracker has gradually become the bottleneck in its further application. Under dynamic conditions, a star spot continuously moves across the image plane of a star tracker during exposure and forms a smeared star spot, which is called a star streak [2,3 ]. Given that its positioning accuracy directly degrades the success rate of star identification, tracking stabilization, and attitude precision, a star streak must be initially located. Thus, two important questions are raised as follows:

To solve these problems, we must first establish an appropriate imaging model for a smeared star spot.

The static star spot imaging model can be expressed as a 2D Gaussian function, which is widely used in the literature [4–6 ]. By contrast, the dynamic star spot imaging model is a complicated integral expression that is related to the motion trajectory of a star spot [7–11 ]. This model has an equivalent form which is represented as the convolution of a motion-blurred point spread function (PSF) and a static star PSF from the perspective of image processing [12–15 ]. However, the integral operation in these two models makes analytical calculations of accuracy estimation difficult. Accordingly, a piecewise model is established for simplicity [16,17 ]. In this model, both ends of a star streak approximate Gaussian profile and its middle section assumes a constant in the direction of motion. This model is suitable for a long star streak but is inaccurate for a shorter one. Dzamba et al. developed a more concise model without integral operation [18,19 ]. However, parameters, such as angular velocity and Gaussian radius, were implied in this model, which had similar difficulty in the analysis of star streak locating accuracy. All these models should perform a 2D integral operation over each pixel as the last step to generate a digital star image, and this process is called pixelization; however, such pixelization is challenging, and none of these models have worked out any analytical expression for a pixel-level image. Thus analytical solutions for succeeding studies still remained problems. To solve this problem, we proposed a concept of the line segment spread function (LSSF) and established a uniform linear motion star spot imaging model based on the LSSF. Although the motion of the star spot in the image plane is non-linear, it is a well approximation to the uniform linear motion under certain conditions which can be easily satisfied. The model was the function of incident flux, exposure time, velocity of a star spot, and Gaussian PSF radius. This model included all the key parameters in imaging process and provided an explicit physical interpretation. Moreover, we carried out the 2D integral in pixelization and provided a pixel-level analytical dynamic star image model.

To obtain high-precision three-axis attitude information, a comprehensive study on accurately determining the location of a smeared star spot is necessary. Given that the centroid method is the most widely used star spot location algorithm, frequency domain analysis has been performed to comprehensively examine systematic errors in this method caused by discrete sampling [6,20,21 ]; however, the conclusion drawn is unsuitable for a smeared star spot. In References [9], [22], and [23], focus was given on the relation between centroiding accuracy and exposure time under dynamic conditions; optimum exposure time was then calculated. In Reference [18], a simulation-based accuracy model was developed by considering motion. The accuracy profile was a function of stellar magnitude in this model. Aretskin-Hariton et al. [24] conducted Monte Carlo simulation to create a 4D lookup table to provide a quick estimation of system capabilities. Based on Cramér–Rao lower bound theorem, Zhang et al. [11] derived the minimum positional error for a moving star spot under non-uniform velocity. However, the conclusions deduced from a 1D smeared star spot were different under 2D conditions. Most of these accuracy estimation models are based on numerical simulation, whereas others have explicit formulas. However, their derivation relatively involves approximations which cause additional errors.

As the only assumption of our dynamic imaging model made is the uniform linear motion of the star spot, we analyzed the centroiding error caused by non-linear motion firstly. Based on our model, we perform an in-depth study of the systematic and random centroiding error of a smeared star spot and derive an analytical centroiding error expression that explicitly shows the effects of incident flux, Gaussian radius, image sensor noise level, star spot velocity, and smear length. Each parameter contributes differently to centroiding error, thus we conduct comprehensive optimization to improve accuracy under dynamic conditions.

2. Dynamic imaging model for a star tracker

Establishing an accurate imaging model for a star tracker is the first step to achieving high star positioning accuracy. The static imaging model assumes a 2D Gaussian function. Under dynamic conditions, a star spot moves across the image plane and forms a streak when a star tracker is rotating. Given that energy distribution remains a Gaussian function at any infinitesimal interval of exposure time, the summation of energy distributions is equal to the integral of a Gaussian function with respect to time. This continuous integral expression performs a 2D integral operation over each pixel to form a digital image. Any detailed research at the pixel level involves calculating the triple integral in the first step. However, such calculation is challenging, and the common method used entails conducting a numerical simulation, which is time consuming. Moreover, analytical solutions for succeeding studies are difficult to obtain. To solve this problem, an analytical energy distribution model for a smeared star spot is established based on LSSF; then, a pixel-level analytical star image model is provided.

If the input of an optoelectronic imaging system is a geometric point, then its output is a dispersive spot, which can be represented by a normalized PSF. The Gaussian function is generally selected as the PSF, and a 1D PSF is represented as

In the imaging process of a star tracker, a star is approximated to a point source at infinity; thus, stellar flux passes through the optical lens and forms a star spot at the focal plane. Under static conditions, the energy distribution of a star spot is expressed as

Under dynamic conditions, the energy distribution of a star spot is the same as the static PSF at any infinitesimal interval of the exposure time, as illustrated in Fig. 1(a) . The movement of a star spot causes it to leave a trail that accumulates in different locations and forms a star streak, which is shown in Figs. 1(b) and 1(c).

 

Fig. 1 Dynamic imaging process of a star spot.

Download Full Size | PDF

In Fig. 1, the projection of the imaging process is plotted in the x direction. t ₀ and t ₃ represent the beginning and the end of exposure time, respectively; t ₀ = 0; t ₃ = T; and [x _c(t), y _c(t)] denote the center coordinates of the star spot in the image plane at time t. A dynamic energy distribution model is expressed as

This integral expression model of a dynamic star spot can describe the star streak caused by any motion. Most previous works have used this model; however, integral expression is difficult to calculate analytically. To solve this problem, we need to make some simplification first. As we know, when a star tracker rotates with a constant angular rate, the star spot moves in the image plane with a changing velocity. Fortunately, if a star tracker’s field of view (FOV) and its rotation angle within exposure is small, the motion of a star spot in the image plane approximates uniform linear motion [7,12,13,16,19 ]. With this approximation, we could derive a dynamic imaging model with an explicit physical interpretation of the imaging. The error due to this approximation is analyzed in the next section.

Let a star spot move with velocity v in the x direction in the image plane. T is the exposure time, L is the smear length, and L = vT. (x ₀, y ₀) and (x ₀ + vt, y ₀) represent the center coordinates of the star at the beginning of exposure and at time t, respectively. According to Eqs. (2) and (4) , the energy distribution model of the smeared star spot is derived as

As shown in Eq. (5), the star streak energy distribution is the function of incident flux, exposure time, Gaussian radius, velocity, and the initial position of the star spot. Based on the comparison between Eq. (5) and Eq. (3), the dynamic star spot imaging model is consistent with the static model while maintaining a similar explicit physical meaning, except that PSF becomes LSSF under dynamic conditions. The star spot dynamic imaging process is equivalent to a static linear light source imaging process, as shown in Fig. 1(c).

Furthermore, let the star spot move with velocity v in the image plane in a straight line, in which the inclination angle is θ. (x ₀, y ₀) and (x ₀ + vt cos θ, y ₀ + vt sin θ) represent the star spot center coordinates at time t ₀ and t, respectively. Then, the star spot energy distribution is derived as

The coordinate system (o–x–y) rotates counterclockwise by an angle θ around its origin into the new coordinate system (o–X–Y), and the initial position (x ₀, y ₀) at the beginning of the exposure transforms into (X ₀, Y ₀) in the new coordinate system. Thus, Eq. (6) is converted into

The rotation transformation is shown in Fig. 2 , and the initial position (x ₀, y ₀) is located at the origin for simplicity.

 

Fig. 2 Rotation transformation of the star streak.

Download Full Size | PDF

Based on the comparison between Eq. (8) and Eq. (5), a star streak in any motion direction can be transformed in the x direction. The profile of the star streak along the motion direction is expressed as the LSSF, and the profile perpendicular to the motion direction is still expressed as the PSF. Moreover, these two profiles are independent from each other, see Fig. 2. By using this model, proving that the intensity of the star streak reaches a peak at the midpoint of the trajectory (x ₀ + vT/2, y ₀) becomes easy, as given by

As shown in Eq. (9), when exposure time is kept constant, E _max is inversely proportional to velocity v, which explains why detection sensitivity decreases under dynamic conditions. If the velocity of a star spot is kept constant when exposure time tends to infinity, then the limiting value of E _max is shown as follows:

Prolonging exposure time under static conditions improves detection performance for dim stars; however, Eq. (10) shows that this method is useless in terms of detection performance under dynamic conditions. When smear length L = 4ρ, E _max = 0.954 E _lim; when L = 6ρ, E _max = 0.997 E _lim. Thus, increasing exposure time minimally contributes to the intensity of the star streak as long as exposure time T ≥ 4ρ/v. Consequently, we define T _e as the effective exposure time and T _e = 4ρ/v. This result is more comprehensive than Liebe’s result [3], which is defined qualitatively as (1 pixel)/v, and more accurate than Wei’s result which involved statistical inductions [22].

The energy distribution of a continuous star streak is sampled by the pixel array of an image sensor and forms a digital star image, as shown in Fig. 3 . A number of photons hit the pixel area and generate a number of electrons; then, the electrons are converted into a voltage signal. Finally, the voltage is amplified and converted into a digital gray value. Symbol η _QE denotes quantum efficiency, which refers to the capability to convert photons into electrons. K denotes conversion gain, which refers to the capability to convert electrons into a digital gray value.

 

Fig. 3 Discrete star streak image.

Download Full Size | PDF

For convenience, the pixel pitch is selected as the length unit. In Fig. 3, (x ₀, y ₀) denotes the initial position of the star spot at the beginning of exposure, and L denotes the smear length. If we define the radius of a star spot as 3ρ, then the star spot collects over 99.7% of the total energy. In practice, only the pixels within the region are included in the calculation. This area is called the centroiding window, which is determined by ρ and L, with length l = 6ρ + L and width w = 6ρ. Suppose (x_i, y_j) are the coordinates of the geometric center of a pixel at the i-th column and the j-th row. The gray value of the pixel is expressed as

Adding the gray value of the pixels of the i-th column yields

The total gray value of a star streak in the image is

Thus, a pixel-level model of the dynamic star image is obtained. The centroiding error can be directly worked out without the Monte Carlo calculations presented in the next section because of the absence of an integral operation.

3. Dynamic star spot centroiding accuracy performance

The most important indicator of a star tracker is three-axis attitude accuracy, and star positioning accuracy is the basis of attitude accuracy. The centroiding algorithm is the most widely used star spot location algorithm. Studies on static star spot centroiding accuracy are thorough and comprehensive; however, dynamic star spot centroiding accuracy has not yet been clearly identified. As the aforementioned dynamic imaging model is derived on the assumption of uniform linear motion, there is an error between the true centroid and the one calculated with our model. So we analyzed this non-linear motion error in the first place. Except for this error, we developed a comprehensive star spot centroiding accuracy model without approximation, and centroiding errors can be classified into two types: systematic errors caused by pixelization and random errors caused by noises. We will analyze these two types of errors individually.

3.1 Centroiding errors caused by non-linear motion

There are two types of rotation. One is the rotation around the boresight [Fig. 1(a)] and the other is the cross-boresight rotation, see Fig. 1(b).

As shown in Fig. 4 , the star tracker based coordinate system is expressed as O-XYZ, f is the focal length, h is the length of a side of the image plane, and the angle between the starlight direction and the boresight direction is θ (θ ∈[0, θ _FOV/2]), where θ _FOV represents the field of view (FOV) of a star tracker.

 

Fig. 4 Two types of rotation. (a) Rotation about boresight; (b) Rotation about cross-boresight axis.

Download Full Size | PDF

In Fig. 4(a), the star tracker rotates about Z axis with an angular rate ω_z and the duration of exposure is t, the rotation angle which is a micro-variation turns to be Δθ_z = ω_z·t, and the smear length of the star streak is expressed as [15]

In Fig. 4(b), the vehicle rotates about an rotation axis which is perpendicular to the boresight axis with an angular rate ${\omega }_{xy}={({\omega }_x^2+{\omega }_y^2)}^{1/2}$, where ω_x and ω_y represent the angular rate about X axis and Y axis respectively, the length of the star streak is given by [12,15 ]

The velocity of the star spot v_xy is derived as

Although Eq. (14) shows that v_z is a constant under a boresight rotation condition, the trajectory of the star spot is a curve. In the situation of cross-boresight rotation, the star spot moves in a straight line, but the velocity v_xy is variable. As we established a dynamic imaging model in the case of uniform linear motion in section 2, we used the constant velocity instead of the true velocity. This makes additional centroiding error caused by non-linear motion. For clarity, we redraw the Fig. 4 as Fig. 5 . In Fig. 5, the initial position of a star spot in the image plane locates at A, when t = 0. Then it moves to B after a time period t.

 

Fig. 5 Centroiding error of the non-linear moion. (a) Boresight rotation; (b) Cross-boresight rotation.

Download Full Size | PDF

In Fig. 5(a), the arc AB is the true trajectory of the star spot, but the trajectory in our model is chord AB. Since the arc and the chord are symmetrical with their midpoint, the centroid of the arc and chord are P and P’ respectively. The centroiding error ε_z caused by boresight rotation is expressed as

Considering the worst case that the exposure t is 0.1s, and the angular rate ω_z is 5°/s. We obtain

For convenience, we choose the pixel pitch as the length unit. The centroiding error ε_z is expressed as

As the maximum error caused by boresight rotation is less than 0.01 pixels. so we concentrate on the major error and neglect the error ε_z caused by boresight rotation.

In Fig. 5(b), we assume point P is the true centroid of the star streak, T is the exposure time. We obtain [7]

If we use the uniform linear motion model, the centroid is expressed as P’. Thus

The centroiding error caused by this approximation is expressed as

As it can be seen from Eq. (22), the error ε_xy is a function of focal length f, cross-boresight velocity ω_xy, exposure time T, and FOV θ _FOV. We found that ε_xy grew larger when f, ω_xy and T increased. And the length of the star streak L_xy is approximate to fω_xyT. So we graphed the error as a function of L_xy under different θ _FOV in Fig. 6 . Long smear length of the star streak is hard to extract and determine its centroid, meanwhile the centroiding error is large, and thus in most cases the smear length is shorter than 30 pixels. So the longest smear length we chose is 30 pixels.

 

Fig. 6 Centroiding error ε_xy vs. smear length.

Download Full Size | PDF

Figure 6 shows that the maximum error caused by the non-linear motion is less than 0.01 pixels which can be neglected. Thus it is reasonable to assume that the star spot moves across the image plane uniformly in a straight line.

3.2 Systematic error in dynamic star spot centroiding

By performing appropriate rotation transformation, a star streak in any motion direction can be transformed into the motion direction parallel to the x-axis. The total centroiding error of this star streak is decomposed into the x-and y-component errors. Although component errors vary in different cases, the total centroiding errors in each case can be proven to be the same. Hence, we concentrate on the case in which the motion direction of a star streak is parallel to the x-axis. By substituting the star streak energy distribution model into the centroiding algorithm, we obtain

By simplifying Eq. (23), the centroids of the x- and y-components are obtained as

Therefore, if we define the midpoint of the trajectory (x ₀ + vT/2, y ₀) as the true position of a star spot (x _c, y _c), then centroid $(\overline{x},\overline{y})$ is the unbiased estimation of the true position. The following research focuses on the centroid of a star steak in the x direction because this star streak does not smear in the y direction.

Given the pixelization, Eq. (23) degrades into

Systematic error in the centroiding algorithm is caused by pixelization. This error is given by

By substituting Eqs. (12) and (13) into Eq. (27), the systematic error of‾x can be written as

As shown in Eq. (28), δ_x is the function of Gaussian radius ρ, smear length L, and initial position x ₀. If x ₀ moves within a pixel, then δ_x changes periodically. When x ₀ is known, this error can be compensated. In practice, however, x ₀ is a random variable that is uniformly distributed over a pixel within the range [−0.5, 0.5). The root mean square error is defined as the systematic error of‾x, as follows:

Systematic error in the centroiding algorithm depends on the size of the centroiding window because the summation of the discrete centroiding algorithm is performed within the window, see Fig. 3. The length and width of the star streak centroiding window are integers; thus, the length and width can be rounded as

 

Fig. 7 σ_x,sys vs. Gaussian radius ρ and smear length L.

Download Full Size | PDF

The length l of the centroiding window is fixed at (9 + ⌈L⌉) in Fig. 7(a) and variable according to Eq. (30) in Fig. 7(b). First, the dependence of systematic error σ_{x ,sys} on Gaussian radius ρ is discussed. The centroid is basically a weighted average of the pixel coordinates, and the weight is the pixel value. When ρ increases, stellar flux spreads to more pixels. Meanwhile, if ρ is considerably smaller than window length, then additional pixel information is used in centroiding calculation. Thus, high centroiding accuracy is acquired and σ_{x ,sys} decreases, as shown in Fig. 7(a).

However, considering the constraints of the centroiding window, considerable stellar flux falls outside the window when ρ is relatively large, and the nonsymmetrical truncation effect increases, which induces more centroiding errors. In summary, increasing ρ improves centroiding accuracy; simultaneously, however, accuracy can be negatively affected by window truncation. When the demerits of the truncation effect overcomes the merits of increasing ρ, σ_{x ,sys} stops decreasing and begins to increase, as illustrated in Fig. 7(b).

Second, the relation between systematic error σ_{x ,sys} and smear length L is discussed. In Fig. 7(a), window length l is considerably longer than the length of the star streak, and the truncation effect of the centroiding window can be ignored. When L < 1 pixel, σ_{x ,sys} decreases with increasing L. When n < L < n + 1 pixels (n is an integer), σ_{x ,sys} reaches the maximum number and then decreases with increasing L. When L = n pixels, σ_{x ,sys} = 0, which is represented as negative infinity under the logarithmic coordinates and is not plotted in the figure for clarity.

In Fig. 7(b), for a smaller ρ relative to l (ρ < 0.42 pixel or 0.5 < ρ < 0.58 pixel), the truncation effect can be ignored; in which case, the errors are similar, as shown in Fig. 7(a). For a larger ρ relative to l, the truncation effect dominates the error. In this case, when n < L < n + 1 pixels, l remains constant, whereas the edge of the star streak approaches the edge of the window with increasing L. Thus, the truncation effect becomes increasingly serious, and σ_{x ,sys} increases. Error function σ_{x ,sys}(ρ,L) is discontinuous because l is an integer. Based on the aforementioned analysis, the overall trend is as follows: when smear length is long, centroiding accuracy is high.

The systematic error of‾y is the limit of σ_{x ,sys} as L approaches zero:

In Figs. 7(a) and 7(b), the black curve in plane L = 0 shows that σ_{y ,sys} is a function of ρ. As shown in the figure, the tendency of σ_{y ,sys} is similar to that of σ_{x ,sys}. If the truncation effect of the centroiding window is ignored, σ_{y ,sys} decreases with increasing ρ. If the width w of the window is constrained to a fixed value, then σ_{y ,sys} reaches a minimum with increasing ρ. Finally, when w alters the optimal parameters, ρ varies accordingly.

σ_{x ,sys} and σ_{y ,sys} are uncorrelated; thus, total systematic error σ _sys can be expressed as

σ _sys is a function of ρ and L, and the graph of Eq. (32) is shown in Fig. 8 .

 

Fig. 8 Total systematic error σ _sys vs. Gaussian radius ρ and smear length L.

Download Full Size | PDF

In Fig. 8(a), a fixed window size (9 + ⌈L⌉) × 9, which is significantly larger than the star streak, is selected. Meanwhile, the window size in Fig. 8(b) is variable according to Eq. (30).

First, the dependence of total systematic error σ _sys on Gaussian radius ρ is discussed. As shown in Fig. 8(a), σ _sys decreases rapidly with increasing ρ. In Fig. 8(b), if window size is fixed, then σ _sys reaches a minimum and increases with increasing ρ.

Second, the relation between total systematic error σ _sys and smear length L is discussed. In Figs. 8(a) and 8(b), the curve of σ_{y ,sys}(ρ) is shifted in plane L = 5, and the curve of σ _sys(ρ, 0.8) is labeled by the red line. When L increases from 0 to 0.8 pixel, σ _sys decreases. When L increases from 0.8 pixel to 5 pixels or longer, σ _sys remains nearly constant, and the maximum bias to σ_{y ,sys} is 0.0024 pixel; hence, when L > 0.8 pixel, total systematic error σ _sys is estimated to be σ_{y ,sys}, σ _sys(ρ, L) ≈σ_{y ,sys}(ρ).

As stated earlier, Gaussian radius ρ and smear length L are mainly responsible for systematic errors in dynamic star spot centroiding. Moreover, when smear length is longer than 0.8 pixel, the total systematic error caused by smear length can be ignored.

3.3 Random error in dynamic star spot centroiding

Noises, which cause random error in dynamic star spot centroiding, are inevitable in the imaging process of an image sensor. We differentiate Eq. (26) with respect to each variable and assume that the random noises of different pixels are uncorrelated with one another. Hence, we derive

The uncertainty of intensity n_I,i is caused by various sources: spatial noises and temporal noises. Spatial noises include photon response non-uniformity, dark signal non-uniformity, and fix-pattern noise. Spatial noises can be removed via careful calibrations of the image sensor; thus, these noises can be ignored. Temporal noises include photon shot noise, dark current noise, readout noise, and quantization noise, with their standard deviations denoted as n _shot, n _dc, n _read, and n _ADC, respectively. Photon shot noise follows a Poisson distribution and is dependent on incident flux; its variance is equal to the counts of photoelectrons in the imaging process. Dark current noise also follows a Poisson distribution, and its variance is equal to the production of dark current and exposure time. The variance of pixel readout noise is equal to 1/(12K ²), where K is the conversion gain. If we define the noises that are independent of the input star signal as additional noises, then dark current noise, readout noise, and quantization noise belong to additional noises, which are expressed as

For intuition, we take the CMV4000 image sensor as an example. The conclusion will not change with other image sensor. The main characters of this sensor are listed in Table 1 . Thus, we have n _add = (13² + 125T + 1/0.076²)^1/2. Exposure time is less than 0.1 s because the update rate of a star tracker is generally higher than 10 Hz under dynamic conditions. Thus, the upper limit of dark current noise is calculated as (125 × 0.1)^1/2 = 3.5 e⁻. If we take the upper limit of dark current noise as 3.5 e⁻, then additional noise n _add is equal to 14 e⁻.

Table 1. Parameters of the CMV4000 Image Sensor

View Table

The variance of the total noise of a pixel is expressed as

As illustrated in Fig. 3, the variance of the noises of the pixels in a column of the centroiding window is expressed as

Substituting Eq. (37) into Eq. (34) yields

The first and the second terms in Eq. (38) are the x-components of the random error caused by photon shot noise and additional noise, respectively. The error caused by photon shot noise σ_{x, shot} is derived as

In general, the integral of the Gaussian function from −3ρ to 3ρ is equal to 0.997, and the LSSF has a similar characteristic; therefore, minor approximation is used in Eq. (39) for simplicity. The error caused by additional noise σ_{x, add} is derived as

Similarly, the y-component of the error caused by photon shot noise σ_{y, shot} is derived as

The y-component of the error caused by additional noise σ_{y, add} is derived as

Thus, total random error in dynamic star spot centroiding is expressed as

As shown in Eq. (43), total random error σ _ran decreases by slowing down the velocity of the star spot, decreasing Gaussian radius, increasing incident flux, and improving quantum efficiency. Moreover, σ _ran increases when smear length is too short or too long.

We concentrate on the dependence of random error on smear length and Gaussian radius. By assuming that v = 100 pixels/s, Ф = 250 × 10³ photons/s, and n _add = 14 e⁻, total random error as a function of smear length under different Gaussian radii can be derived, as shown in Fig. 9 .

 

Fig. 9 Total random error vs. smear length under different Gaussian radii.

Download Full Size | PDF

As shown in Fig. 9, total random error reaches the minimum number and then increases gradually. In this phenomenon, the increase in star streak energy rises faster than the increase in noises when smear length begins to increase from zero; thus, when the star streak is bright, centroid estimation is accurate. However, star streak energy rarely increases when smear length exceeds a certain value, whereas noises still increase steadily. Thus, centroiding error begins to increase. Another conclusion can be drawn from Fig. 9, i.e., total random error increases when Gaussian radius increases. The increase in Gaussian radius indicates that the star streak extends to more pixels, and the centroiding window enlarges accordingly. Given that more noises are included in this process, random errors increase.

Total centroiding error is determined by both systematic and random errors and is expressed as

4. Parameter optimization

The theoretical analysis of systematic and random errors in the centroiding algorithm is presented in the previous section, and the analytical expression is deduced. Centroiding error under dynamic conditions is the function of Gaussian radius ρ, smear length L, star spot velocity v, noise level n _add, and incident flux Φ. The function is a monotone function with respect to v, n _add, and Φ; and centroiding accuracy improves with decreasing n _add and increasing v and Φ. For ρ and L, however, a value that is either too large or too small cannot improve centroiding accuracy. Therefore, optimizing these two parameters is necessary to obtain an optimal performance of a star tracker. Smear length comprises star spot velocity and exposure time, and exposure time is easier to control. Therefore, we attempt to achieve the optimal exposure time by optimizing smear length.

The hardware configuration of the star tracker is given before optimization. It should be noted that the derivation and conclusion is suit for most of the star trackers. The parameters of the optical lens are listed as follows: aperture D = 40 mm, FOV θ _FOV = 14.5°, focal length f = 44.16 mm, and optical transmission τ = 0.85. Therefore, we substitute incident flux and star spot velocity to star visual magnitude m and star tracker angular velocity ω, respectively. Then, the incident flux of a star of visual magnitude m is expressed in photon counts per second, as given by [1]

For a given angular rate ω, a pure cross-boresight rotation provides the maximum smear length of a star streak which is about 7.5 times longer than the smear length caused by a rotation about boresight [19]. So we constrained the rotation as cross-boresight rotation. From Eq. (16), star spot velocity v is approximately proportional to cross-boresight angular velocity, i.e.,

4.1 Exposure time optimization

The smearing effect is the main factor that decreases the dynamic performance of a star tracker. Based on the previous conclusion, systematic error decreases rapidly and then nearly remains constant, whereas random error is reduced to its minimum number and then increases with increasing smear length L. The optimization of L should consider both systematic and random errors. This optimization can be simplified when L > 0.8 pixel because systematic error is considered a constant in this case. As discussed in Section 2, exposure time should be longer than effective exposure time T _e to ensure that the star streak can be detected; thus, L is longer than Gaussian PSF radius ρ by four times. Moreover, ρ is always greater than 0.2 pixel for a star tracker; hence, L > 0.8 pixel is satisfied.

Based on Eqs. (39) and (41) , random error caused by photon shot noise σ _shot is derived as

If σ _shot reaches its minimum, then optimal L is obtained. By deriving function σ _shot(L) with respect to L and letting the derived function be equal to zero, then the equation yields

Based on Eqs. (40) and (42) , random error caused by additional noise n _add is derived as

Similar to the solution to L ₁, the upper limit of the optimal smear length is derived as

Suppose smear length ρ = 0.5 pixel, additional noise n _add = 14 e⁻, angular velocity ω = 1°/s, and stellar magnitude m = 5. Random error in centroiding is shown in Fig. 10 .

 

Fig. 10 random error of centroiding vs. smear length.

Download Full Size | PDF

As shown in Fig. 10(a), optimal smear length L _o is between L ₁ and L ₂, and L _o is obtained by

L _o varies between L ₁ and L ₂ when m, ω, and n _add change. The dependence of random error as a function of smear length with different m, ω, and n _add values is shown in Figs. 10(b), 7(c), and 7(d) , respectively. Symbol * indicates the optimal smear length of the specific cases.

As shown in Fig. 10, random error σ _ran increases and optimal smear length L _o approaches its upper limit L ₂ when the incident stellar magnitude, star tracker angular velocity, and additional noise of the image sensor are increased.

Optimal exposure time T _o is determined by T _o = L _o/v, where v is calculated by Eq. (46). The lower limit T ₁ and upper limit T ₂ of exposure time are determined by T ₁ = max{T _e, L ₁/v} and T ₂ = max{1/n _fps, L ₂/v}, where n _fps indicates frame per second.

Gaussian PSF radius and additional noises are determined by the hardware configuration of the star tracker. By assuming that n _add = 14 e⁻ and ρ = 0.5 pixel, the optimal exposure time as a function of the angular velocity of a star tracker under different incident stellar magnitudes is illustrated in Fig. 11 .

 

Fig. 11 Optimal exposure time vs. angular velocity under different stellar magnitudes.

Download Full Size | PDF

As shown in Fig. 11, optimal exposure time T _o is between T ₁ and T ₂. If angular velocity or incident stellar magnitude decreases, then T _o approaches the lower limit of exposure. By contrast, if angular velocity or incident stellar magnitude increases, then T _o approaches the upper limit of exposure.

4.2 Gaussian PSF radius optimization

Stellar rays can be approximated to parallel light rays. The rays pass through the optical lens and are focused on a point in the focal plane. However, the lens is generally slightly defocused to improve the positioning accuracy of a star spot, which spreads to several pixels in the image plane. The profile of a star spot can be described by Gaussian PSF and LSSF, and the parameter Gaussian PSF radius ρ indicates the extent of dispersion. When ρ is high, the region where a star spot spreads out is large, and the centroiding window is proportional to ρ. Systematic error reaches its minimum number and then increases with increasing ρ, whereas random error increases with increasing ρ; hence, an optimal Gaussian radius is used to minimize centroiding error.

Optimal smear length L _o is longer than 0.8 pixel, which is deduced from Eq. (51); therefore, systematic error is simplified as a function of one variable, σ _sys(ρ, L _o) ≈σ_{y ,sys}(ρ). If incident stellar magnitude m, angular velocity ω, and additional noises n _add are determined, and smear length is taken as L _o, then random error σ _ran is also simplified as a simple function of one variable. Thus, total centroiding error σ _c is a function of Gaussian PSF radius ρ, and optimal Gaussian PSF radius ρ _o is derived as

Total centroiding error reaches its minimum number when ρ is equal to ρ _o. σ _sys, σ _ran, and σ _c are illustrated in Fig. 12(a) , where m = 5, ω = 1°/s, and n _add = 14 e⁻, and the size of the centroiding window is obtained using Eq. (30). The relation between total centroiding error and Gaussian radius with different m, ω, and n _add values is shown in Figs. 12(b), 12(c), and 12(d), respectively. Symbol * indicates the optimal Gaussian radius.

 

Fig. 12 Centroiding error vs. Gaussian radius.

Download Full Size | PDF

As shown in Fig. 12(a), the systematic error, the random error and the total error in centroiding are the piecewise function of the centroiding window size. When the window is (3 + ⌈L _o⌉) × 3, random error increases from 0.024 pixel to 0.025 pixel with increasing ρ, systematic error reaches its minimum value of 0.011 pixel and then increases to 0.018 pixel, and total error reaches its minimum value of 0.027 pixel and then increases to 0.031 pixel. When the window is (5 + ⌈L _o⌉) × 5 or (7 + ⌈L _o⌉) × 7, random error is considerably larger than systematic error; thus, random error, which is approximated to be the total error, reaches its minimum value of 0.038 pixel at ρ = 0.5 pixel. Optimal Gaussian radius ρ _o is equal to 0.42 pixel, and the best window size is (3 + ⌈L _o⌉) × 3.

As shown in Figs. 12(b), 12(c), and 12(d), total error increases with increasing stellar magnitude, angular velocity, and additional noises. However, the increment differs among various window sizes; the optimal Gaussian radius is 0.5 pixel, and the best window size is (5 + ⌈L _o⌉) × 5 when m < 3.8 or ω < 0.3°/s or n _add < 3.3 e⁻. In addition, the optimal Gaussian radius is 0.42 pixel, and the best window size is (3 + ⌈L _o⌉) × 3 in the other cases.

5. Simulations and night sky experiment

In the previous sections, we discuss how we have developed a dynamic imaging model of a star tracker and analyzed systematic and random centroiding errors. The analytical expression of the total error has been derived, and smear length and Gaussian radius have been optimized. To validate the imaging model, centroiding error expression, and optimal parameters, we have conducted numerical simulations and a night sky experiment. The numerical simulations have focused on the accuracy of the results, whereas the night sky experiment has provided the final validation of correctness.

5.1 Numerical simulations

The simulation conditions are set as follows. Angular velocity ω = 1°/s, and incident stellar magnitude m = 5. The image sensor used is CMV4000, the parameters of which are listed in Table 1. The parameters of the optical lens are the same as those in the previous section. Gaussian PSF radius ρ is set as 0.70 and 0.42 pixel in Fig. 13 and Fig. 14 , respectively.

 

Fig. 13 Comparison of centroiding errors of different methods.

Download Full Size | PDF

 

Fig. 14 Centroiding error vs. smear length L.

Download Full Size | PDF

Numerical simulation is performed through the following procedure. First, we generate 150 groups of star streaks using the dynamic imaging model. Smear length ranges from 0.1 pixel to 15 pixels, with a step of 0.1 pixel. Each group contains 121 star streaks. The initial position of each star streak is uniformly distributed within a single pixel, and then photon shot noise and additional noises are added. Second, we calculate each centroid of the 121 star streaks using the centroiding algorithm. The absolute centroiding error is easily derived because the true centroid is known. Then, the standard deviation of the errors in one group is considered the centroiding error at a certain smear length, which is shown as a red scattered point. Finally, we repeat this procedure until all groups of star streaks are processed.

The theoretical centroiding error from Eq. (35) is denoted in Fig. 13 with a solid line. The centroiding error of Wei’s method is denoted with a dash line, and the simulation is denoted with scattered points.

As shown in Fig. 13, the scattered points spread around the theoretical value of ours, but there is a deviation in Wei’s [22] results from the simulation. Thus our analysis is more accuracy than Wei’s.

The theoretical centroiding error from Eq. (44) is denoted in Fig. 14 with a red solid line. The relation between centroiding error and smear length with different angular velocities and incident stellar magnitudes are illustrated in Figs. 14(a) and 14(b), respectively.

As shown in Fig. 14, the scattered points spread around the theoretical value. Moreover, the trend of the simulation points is the same as the theoretical trend. Both points reach the minimum number rapidly and then increase with increasing smear length. Therefore, the simulations agree well with the theoretical results.

The centroiding error reaches its minimum number when smear length is equal to L _o, as shown by the green square points in Fig. 14. The optimal smear length derived from the simulation highly approximates the theoretical optimal value L _o; thus, the optimization of smear length is correct and precise. When angular velocities are 0.5°/s, 1°/s, and 2°/s, the optimal smear lengths are 3.60, 4.28, and 5.03 pixels, respectively. In addition, when stellar magnitudes are 4, 5, and 6, the optimal smear lengths are 3.40, 4.28, and 5.28 pixels, respectively.

5.2 Night sky experiment

For the final validation of the imaging model, centroiding error expression, and optimal parameters, a night sky experiment was conducted at the Xinglong Station (National Astronomical Observatories, China). The hardware configurations of the star tracker were the same as those previously mentioned. In Fig. 15 , the star tracker was mounted on a turntable that was installed on a tripod, and the boresight of the star tracker was perpendicular to the rotation axis; thus, the cross-boresight angular velocity of the star tracker was equal to the slew rate of the turntable, which was set at 1°/s and 2°/s. Exposure time was set at 10, 20, 30, 40, 50, 60, 80, and 100 ms. Smear length varied from 1.41 pixels to 14.12 pixels when long star streaks were discarded because a long smear length was meaningless. Over 250 frames of centroid data were acquired in each of the 12 cases (with 3 cases discarded). Finally, the centroid data were transferred to a PC for recording.

 

Fig. 15 Setup of the night sky experiment.

Download Full Size | PDF

Although the true positions of the star streaks are unavailable, the relative position of two stars is accurate because of the invariance of inter-star angles. Therefore, we selected star angular distance as the criterion to evaluate centroiding error. The evaluation was similar to analyzing noise equivalent angle error. The centroiding error of a single star streak was obtained by selecting a pair of identified stars, calculating the angular distance of these two star vectors, comparing the obtained distance with the one from the catalog, dividing the difference by 2^1/2, and converting its unit into pixels. Statistics on the average centroiding error and its deviation in all frames of each case were calculated. Finally, the standard deviations for different cases were plotted in Fig. 16 .

 

Fig. 16 Centroiding error vs. smear length under different velocities in the night sky experiment.

Download Full Size | PDF

In Fig. 16, centroiding error decreases rapidly and then increases gradually with increasing smear length; this trend is consistent with the previous conclusions. When the slew rate of the star tracker is 1°/s, the optimal smear length is 4.24 pixels. At a slew rate of 2°/s, centroiding error increases dramatically, and the optimal smear length increases to 5.65 pixels. In contrast to the numerical results, i.e., the optimal smear lengths are 4.28 pixels and 5.03 pixels at the slew rates of 1°/s and 2°/s, respectively, the optimal smear lengths from the experiment agree with the theoretical values. Thus, the optimization of the smear length is validated. However, centroiding errors are larger than the theoretical value. This phenomenon may be attributed to atmospheric turbulence which is beyond the scope of this study, and thus, are not considered in the model. In summary, the night sky experiment validates the reliability of our error analysis and parameter optimization.

6. Conclusion

In this study, we modeled the dynamic imaging of a star tracker. With the assumption of uniform linear motion of a star spot, a dynamic star spot was expressed as the product of incident flux, exposure time, PSF, and LSSF because the PSF of a static star spot is expressed as a Gaussian function. As the centroiding error caused by non-linear motion is less than 0.01 pixels even in worst cases, the dynamic imaging model we established is applicable and accurate. Moreover, we developed a pixel-level image model of a star streak.

Since the centroiding error caused by non-linear motion is neglected, the continuous-form centroiding algorithm is an unbiased method for estimating the center of a star streak. Pixel sampling causes systematic error, whereas noises in the imaging process cause random error. We analyzed these two types of errors and derived the analytical expressions for each type. Consequently, we determined that the systematic centroiding error component perpendicular to motion direction was the same as the static star spot centroiding error, and the error in motion direction was relative to smear length. Furthermore, systematic error is the function of Gaussian PSF radius ρ, smear length L, and the size of the centroiding window l × w, whereas random error is the function of ρ, L, incident flux Φ, quantum efficiency η _QE, noise level n _add, and window size. Systematic error decreases with increasing window size, reaches its minimum number, and then increases with increasing ρ and decreases with increasing L. If L is longer than 0.8 pixel, then systematic error approximates a constant. By contrast, random error decreases with increasing ρ, reaches its minimum number, and then increases gradually with increasing L. In addition, random error decreases with increasing Φ, improving η _QE, and decreasing n _add. The total centroiding error will reach its minimum when the optimal parameters are considered. When the incident star is dim, the slew rate is fast, and the noise level is high, the optimal window width is 3 pixels, the optimal smear length varies from 2 pixels to 7 pixels, and optimal ρ is 0.42 pixels.

The results of the numerical simulations agree with the theoretical results; hence, the night sky experiment validates the conclusion of this study.