Abstract

Under dynamic conditions, star spots move across the image plane of a star tracker and form a smeared star image. This smearing effect increases errors in star position estimation and degrades attitude accuracy. First, an analytical energy distribution model of a smeared star spot is established based on a line segment spread function because the dynamic imaging process of a star tracker is equivalent to the static imaging process of linear light sources. The proposed model, which has a clear physical meaning, explicitly reflects the key parameters of the imaging process, including incident flux, exposure time, velocity of a star spot in an image plane, and Gaussian radius. Furthermore, an analytical expression of the centroiding error of the smeared star spot is derived using the proposed model. An accurate and comprehensive evaluation of centroiding accuracy is obtained based on the expression. Moreover, analytical solutions of the optimal parameters are derived to achieve the best performance in centroid estimation. Finally, we perform numerical simulations and a night sky experiment to validate the correctness of the dynamic imaging model, the centroiding error expression, and the optimal parameters.

© 2016 Optical Society of America

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:

  • (1) How does smearing affect positioning accuracy?
  • (2) How can the positioning error of a star streak be reduced?

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

fPSF(x)=12πρexp(x22ρ2),
where ρ indicates the Gaussian PSF radius, which expresses the spread scale of the optical lens. If the input is a straight line, then the output is an LSF, which is the integral of the PSF on the 1D coordinate. If the input is a line segment, we derive that the output is the definite integral of the PSF from one endpoint of the line segment to another, which is denoted as the LSSF in this study:
fLSSF(x)=1LxLxfPSF(u)du=12L[erf(x2ρ)erf(xL2ρ)],
where L is the projection length of the line segment on the image plane, and x-L and x are the coordinates of the endpoints of the line segment in the image space.

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

Esta(x,y)=ΦT2πρ2exp[(xxc)2+(yyc)22ρ2]=ΦTfPSF(xxc)fPSF(yyc),
where Ф is the incident flux of the star on the image plane, T is the exposure time, and (x c, y c) represent the coordinates of the center position of the star spot.

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).

 figure: Fig. 1

Fig. 1 Dynamic imaging process of a star spot.

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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

Edyn(x,y)=0TΦ2πρ2exp{[xxc(t)]2+[yyc(t)]22ρ2}dt=Φ0TfPSF(xxc(t))fPSF(yyc(t))dt.

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

Edyn(x,y)=Φ0TfPSF(xx0vt)fPSF(yy0)dt=Φ2v[erf(xx02ρ)erf(xx0vT2ρ)]fPSF(yy0)=ΦTfLSSF(xx0)fPSF(yy0).

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

Edyn(x,y)=Φ0TfPSF(xx0vtcosθ)fPSF(yy0vtsinθ)dt=Φ2v[erf(u2ρ)erf(uvT2ρ)]12πρexp(v22ρ2)=ΦTfLSSF(u)fPSF(v),
where u=(xx0)cosθ+(yy0)sinθ and v=(xx0)sinθ+(yy0)cosθ. Let

{x=XcosθYsinθy=Xsinθ+Ycosθ.

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

Edyn(X,Y)=ΦTfLSSF(XX0)fPSF(YY0).

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

 figure: Fig. 2

Fig. 2 Rotation transformation of the star streak.

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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

Emax(X,Y)=ΦTfLSSF(vT/2)fPSF(y0)=Φ2πρverf(vT22ρ).

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:

Elim=limTΦ2πρverf(vT22ρ)=Φ2πρv.

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.

 figure: Fig. 3

Fig. 3 Discrete star streak image.

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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 (xi, yj) 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

Iij=yj0.5yj+0.5xi0.5xi+0.5ηQEKEdyn(x,y)dxdy=ΦTηQEKxi0.5xi+0.5fLSSF(xx0)dxyj0.5yj+0.5fPSF(yy0)dy,=ΦTηQEKFLSSF(xix0)FPSF(yjy0)
where FPSF(yj)=yj0.5yj+0.5fPSF(y)dy=12{erf[(yi+0.5)/(2ρ)]erf[(yi0.5)/(2ρ)]},

FLSSF(xi)=xi0.5xi+0.5fLSSF(x)dx=12L{(xi+0.5)erf[(xi+0.5)/(2ρ)]+2ρ2fPSF(xi+0.5)(xi0.5)erf[(xi0.5)/(2ρ)]2ρ2fPSF(xi0.5)(xi+0.5L)erf[(xi+0.5L)/(2ρ)]2ρ2fPSF(xi+0.5L)(xi0.5L)erf[(xi0.5L)/(2ρ)]+2ρ2fPSF(xi0.5L)}.

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

Ii=jIij=ΦTηQEKFLSSF(xix0).

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

Itotal=ijIij=iIi=ΦTηQEK.

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.

 figure: Fig. 4

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

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

Lz(t)=ftanθΔθz=ftanθωzt=vzt,
where v z is the linear velocity of the star spot in the image plane in the case of boresight rotation.

In Fig. 4(b), the vehicle rotates about an rotation axis which is perpendicular to the boresight axis with an angular rate ωxy=(ωx2+ωy2)1/2, where ωx and ωy represent the angular rate about X axis and Y axis respectively, the length of the star streak is given by [12,15 ]

Lxy(t)=f[tan(θ+Δθxy)tanθ]=f[tan(θ+ωxyt)tanθ].

The velocity of the star spot vxy is derived as

vxy(t)=dLxy(t)dt=fωxy[1+tan2(θ+ωxyt)]fωxy.

Although Eq. (14) shows that vz 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 vxy 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.

 figure: Fig. 5

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

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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 εz caused by boresight rotation is expressed as

εz=|P'P|=|OP||OP'|=r(1cosωzt2).

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

εz=r(1cosωzt2)h2(1cosωzt2)=4.76×106h.

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

εz4.76×106h/d=0.0097pixels,
where d represents the pixel pitch, and h/d represents the number of pixels in a side of the image sensor which is assumed as a square and h/d = 2048.

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 εz 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]

|AP|=1T0TLxy(t)dt=fωxyTlncosθcos(θ+ωxyT)ftanθ,

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

|AP'|=vxy(0)T2=fωxy[1+tan2(θ)]T2.

The centroiding error caused by this approximation is expressed as

εxy=|AP||AP'|=fωxyTlncosθcos(θ+ωxyT)ftanθfωxy[1+tan2(θ)]T2fωxyTlncos(θFOV/2)cos(θFOV/2+ωxyT)ftanθFOV2fωxy[1+tan2θFOV2]T2

As it can be seen from Eq. (22), the error εxy is a function of focal length f, cross-boresight velocity ωxy, exposure time T, and FOV θ FOV. We found that εxy grew larger when f, ωxy and T increased. And the length of the star streak Lxy is approximate to xyT. So we graphed the error as a function of Lxy 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.

 figure: Fig. 6

Fig. 6 Centroiding error εxy vs. smear length.

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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

x¯=xEdyn(x,y)dxdyEdyn(x,y)dxdy,y¯=yEdyn(x,y)dxdyEdyn(x,y)dxdy.

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

x¯=xfLSSF(xx0)dx=x0+vT/2=xc.y¯=yfPSF(yy0)dy=y0=yc

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 (x¯,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

x¯=ijxiIijijIij,
where summation is performed within the centroiding window. By substituting Eq. (12) into Eq. (25), the 1D discrete centroiding algorithm is obtained:

x¯=ixiIiiIi.

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

δx=x¯xc=ixiIiiIi(x0+L2).

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

δx=ixiΦTηQEKFLSSF(xix0)ΦTηQEK(x0+L2)=12Lixixi0.5xi+0.5[erf(xx02ρ)erf(xx0L2ρ)]dx(x0+L2).

As shown in Eq. (28), δx is the function of Gaussian radius ρ, smear length L, and initial position x 0. If x 0 moves within a pixel, then δx 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:

σx,sys(σ,L)=[0.50.5δx2(ρ,L,x0)dx0]1/2.

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

l=2×3ρ0.5+1+L,w=2×3ρ0.5+1,
where ⌈∙⌉ denotes the integer roundup operation. When 6ρ ≤ 3 pixels, window size l × w is (3 + ⌈L⌉) × 3; when 3 < 6ρ ≤ 5 pixels, window size is (5 + ⌈L⌉) × 5; when 5 < 6ρ ≤ 7 pixels, window size is (7 + ⌈L⌉) × 7. Under the constraints of window size, the x-component of systematic error σx ,sys serves as a function of Gaussian radius ρ and smear length L, as shown in Fig. 7 .

 figure: Fig. 7

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

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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 σx ,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 σx ,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, σx ,sys stops decreasing and begins to increase, as illustrated in Fig. 7(b).

Second, the relation between systematic error σx ,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, σx ,sys decreases with increasing L. When n < L < n + 1 pixels (n is an integer), σx ,sys reaches the maximum number and then decreases with increasing L. When L = n pixels, σx ,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 σx ,sys increases. Error function σx ,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 σx ,sys as L approaches zero:

σy,sys(ρ)=limL0σx,sys(ρ,L).

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

σx ,sys and σy ,sys are uncorrelated; thus, total systematic error σ sys can be expressed as

σsys=σx,sys2+σy,sys2.

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

 figure: Fig. 8

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

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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 σy ,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 σy ,sys is 0.0024 pixel; hence, when L > 0.8 pixel, total systematic error σ sys is estimated to be σy ,sys, σ sys(ρ, L) ≈σy ,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

σx,ran2=i(x¯xi)2nx,i2+i(x¯Ii)2nI,i2=i(IiItoltal)2nx,i2+i(xix¯Itoltal)2nI,i2,
where nx,i denotes the uncertainty of the geometric center of a pixel, which is determined by the manufacturing technique of the image sensor, and nI,i denotes the uncertainty of the intensity of a pixel. The first and the second terms in Eq. (33) are the centroiding errors caused by nx,i and nI,i, respectively. The first term is ignored because nx,I is small; thus, random error in the centroiding algorithm can be written as

σx,ran2=i(xix¯Itoltal)2nI,i2.

The uncertainty of intensity nI,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

nadd2=ndc2+nread2+nADC2=IdarkT+nread2+1/(12K2).

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.

Tables Icon

Table 1. Parameters of the CMV4000 Image Sensor

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

nI,ij2=K2(nij,shot2+nadd2)=KIij+K2nadd2.

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

nI,i2=jnI,ij2=KIi+wK2nadd2.

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

σx,ran2=KItoltal2i(xix¯)2Ii+wK2nadd2Itoltal2i(xix¯)2=1ΦTηQE3ρ+x0x0+L+3ρ(xix0L/2)2fLSSF(xi)dxi+wnadd2(ΦTηQE)23ρ+x0x0+L+3ρ(xix0L/2)2dxi.

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 σx, shot is derived as

σx,shot2=1ΦTηQE3ρ+x0x0+L+3ρ(xix0L/2)2fLSSF(xi)dxi1ΦTηQE(xiL/2)2fLSSF(xi)dxi=vΦηQEL(112L2+ρ2).

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 σx, add is derived as

σx,add2=wnadd2(ΦTηQE)23ρ+x0x0+L+3ρ(xix0L/2)2dxi=6ρv2nadd2Φ2ηQE2L2112(L+6ρ)3.

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

σy,shot2=KItoltal2j(yjy¯)2Ij=vρ2ΦηQEL.

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

σy,add2=K2nadd2lItoltal2j(yjy¯)2=18v2nadd2(L+6ρ)ρ3(ΦηQEL)2.

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

σran2=(σx,shot2+σy,shot2)+(σx,add2+σy,add2)=vΦηQEL(112L2+2ρ2)+v2nadd2ρ(L+6ρ)2(ΦηQEL)2(L2+12Lρ+72ρ2).

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 .

 figure: Fig. 9

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

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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

σc2=σsys2+σran2.

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]

Φ=τE0/ε2.512mπD2/4,
where m is the star visual magnitude, E 0 is the irradiance of an m = 0 star of spectral class G2 at the surface of the Earth’s atmosphere, E 0 = 2.96 × 10−14 W/mm2 [1], and ε is the average energy of a photon. Thus, the star tracker with the aforementioned configurations generates 223 × 103 photons/s for a star of m = 6, Ф = 223 × 103 photons/s.

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.,

vfω=h/2dtan(θFOV/2)ω=Npix/2tan(θFOV/2)ω,
where N pix is the square root of the number of active pixels (Table 1). Based on Eq. (46), when ω = 1°/s, v ≈141 pixels/s.

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

σshot2=σx,shot2+σy,shot2=vΦηQEL(112L2+2ρ2).

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

L1=26ρ4.899ρ,
where L 1 represents the lower limit of the optimal smear length. Zhang et al. [11] derived the minimum locating error and reported that the error from motion would dominate the process when smear length was beyond 2 × 31/2 ρ, however this conclusion deduced from a 1D smeared star spot should be generated to the 2D case which is exactly showed in Eq. (48).

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

σadd2=σx,add2+σy,add2=18v2nadd2ρ(L+6ρ)(ΦηQEL)2(L2+12Lρ+72ρ2).

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

L2=2ρ[(54+633)1/3+12(54+633)1/3]14.289ρ.

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 .

 figure: Fig. 10

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

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

Lo=argL{min[σran2(L)]},L[L1,L2].

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 .

 figure: Fig. 11

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

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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) ≈σy ,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

ρo=argρ{min[σc2(ρ)]}=argρ{min[σsys2(ρ)+σran2(ρ)]}.

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.

 figure: Fig. 12

Fig. 12 Centroiding error vs. Gaussian radius.

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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.

 figure: Fig. 13

Fig. 13 Comparison of centroiding errors of different methods.

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 figure: Fig. 14

Fig. 14 Centroiding error vs. smear length L.

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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.

 figure: Fig. 15

Fig. 15 Setup of the night sky experiment.

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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 21/2, 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 .

 figure: Fig. 16

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

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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.

Acknowledgments

This work was supported by National Natural Science Fund of China (NSFC) (No. 61222304) and Specialized Research Fund for the Doctoral Program of Higher Education of China (No. 20121102110032). We gratefully acknowledge the supports.

References and links

1. C. C. Liebe, “Accuracy performance of star trackers-a tutorial,” IEEE Trans. Aerosp. Electron. Syst. 38(2), 587–599 (2002). [CrossRef]  

2. M. A. Samaan, T. C. Pollock, and J. L. Junkins, “Predictive centroiding for star trackers with the effect of image smear,” J. Astronaut. Sci. 50(1), 113–123 (2002).

3. C. C. Liebe, K. Gromov, and D. M. Meller, “Toward a stellar gyroscope for spacecraft attitude determination,” J. Guid. Control Dyn. 27(1), 91–99 (2004). [CrossRef]  

4. G. Rufino and D. Accardo, “Enhancement of the centroiding algorithm for star tracker measure refinement,” Acta Astronaut. 53(2), 135–147 (2003). [CrossRef]  

5. B. R. Hancock, R. C. Stirbl, T. J. Cunningham, B. Pain, C. J. Wrigley, and P. G. Ringold, “CMOS active pixel sensor specific performance effects on star tracker/imager position accuracy”, in Symposium on Integrated Optics, (International Society for Optics and Photonics, 2001), pp. 43–53. [CrossRef]  

6. H. Jia, J. Yang, X. Li, J. Yang, M. Yang, Y. Liu, and Y. Hao, “Systematic error analysis and compensation for high accuracy star centroid estimation of star tracker,” Sci. China Technol. Sci. 53(11), 3145–3152 (2010). [CrossRef]  

7. J. Shen, G. Zhang, and X. Wei, “Simulation analysis of dynamic working performance for star trackers,” J. Opt. Soc. Am. A 27(12), 2638–2647 (2010). [CrossRef]   [PubMed]  

8. H. Wang, W. Zhou, X. Cheng, and H. Lin, “Image smearing modeling and verification for strapdown star sensor,” Chin. J. Aeronauti. 25(1), 115–123 (2012). [CrossRef]  

9. X. Li and H. Zhao, “Analysis of star image centroid accuracy of an APS star sensor in rotation,” Aerospace Control Appl. 35(4), 11–16 (2009).

10. S. Zhao, Y. Wang, H. Wang, and C. Ji, “Study on dynamic angle-measurement compensation for star sensor,” in Natural Computation (ICNC),2011Seventh International Conference on, (IEEE, 2011), 255–258. [CrossRef]  

11. Z. Jun, H. Yuncai, W. Li, and L. Da, “Studies on dynamic motion compensation and positioning accuracy on star tracker,” Appl. Opt. 54(28), 8417–8424 (2015). [CrossRef]   [PubMed]  

12. X. Wu and X. Wang, “Multiple blur of star image and the restoration under dynamic conditions,” Acta Astronaut. 68(11–12), 1903–1913 (2011).

13. T. Sun, F. Xing, Z. You, X. Wang, and B. Li, “Smearing model and restoration of star image under conditions of variable angular velocity and long exposure time,” Opt. Express 22(5), 6009–6024 (2014). [CrossRef]   [PubMed]  

14. W. Zhang, W. Quan, and L. Guo, “Blurred star image processing for star sensors under dynamic conditions,” Sensors (Basel) 12(12), 6712–6726 (2012). [CrossRef]   [PubMed]  

15. C. S. Liu, L. H. Hu, G. B. Liu, B. Yang, and A. J. Li, “Kinematic model for the space-variant image motion of star sensors under dynamical conditions,” Opt. Eng. 54(6), 063104 (2015). [CrossRef]  

16. W. Hou, H. Liu, Z. Lei, Q. Yu, X. Liu, and J. Dong, “Smeared star spot location estimation using directional integral method,” Appl. Opt. 53(10), 2073–2086 (2014). [CrossRef]   [PubMed]  

17. C. A. Nardell, J. Wertz, and P. B. Hays, “Image processing, simulation and performance predictions for the MicroMak star tracker,” in Optics & Photonics 2005, (International Society for Optics and Photonics, 2005), 59160U–59160U–59112.

18. T. Dzamba and J. Enright, “Optical trades for evolving a small arcsecond star tracker,” in Aerospace Conference,2013IEEE, (IEEE, 2013), pp. 1–9. [CrossRef]  

19. T. Dzamba and J. Enright, “Ground testing strategies for verifying the slew rate tolerance of star trackers,” Sensors (Basel) 14(3), 3939–3964 (2014). [CrossRef]   [PubMed]  

20. X. Wei, J. Xu, J. Li, J. Yan, and G. Zhang, “S-curve centroiding error correction for star sensor,” Acta Astronaut. 99, 231–241 (2014). [CrossRef]  

21. J. Yang, B. Liang, T. Zhang, and J. Song, “A novel systematic error compensation algorithm based on least squares support vector regression for star sensor image centroid estimation,” Sensors (Basel) 11(12), 7341–7363 (2011). [CrossRef]   [PubMed]  

22. X. Wei, W. Tan, J. Li, and G. Zhang, “Exposure Time Optimization for Highly Dynamic Star Trackers,” Sensors (Basel) 14(3), 4914–4931 (2014). [CrossRef]   [PubMed]  

23. B. Shen, J. Tan, J. Yang, and J. Liao, “Exposure Time Optimization of the Star Sensor,” Opto-Electron. Eng. 12, 008 (2009).

24. E. Aretskin-Hariton and A. J. Swank, “Star Tracker Performance Estimate with IMU,” in AIAA Guidance, Navigation, and Control Conference (American Institute of Aeronautics and Astronautics, 2015), pp 0334.

References

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  • |

  1. C. C. Liebe, “Accuracy performance of star trackers-a tutorial,” IEEE Trans. Aerosp. Electron. Syst. 38(2), 587–599 (2002).
    [Crossref]
  2. M. A. Samaan, T. C. Pollock, and J. L. Junkins, “Predictive centroiding for star trackers with the effect of image smear,” J. Astronaut. Sci. 50(1), 113–123 (2002).
  3. C. C. Liebe, K. Gromov, and D. M. Meller, “Toward a stellar gyroscope for spacecraft attitude determination,” J. Guid. Control Dyn. 27(1), 91–99 (2004).
    [Crossref]
  4. G. Rufino and D. Accardo, “Enhancement of the centroiding algorithm for star tracker measure refinement,” Acta Astronaut. 53(2), 135–147 (2003).
    [Crossref]
  5. B. R. Hancock, R. C. Stirbl, T. J. Cunningham, B. Pain, C. J. Wrigley, and P. G. Ringold, “CMOS active pixel sensor specific performance effects on star tracker/imager position accuracy”, in Symposium on Integrated Optics, (International Society for Optics and Photonics, 2001), pp. 43–53.
    [Crossref]
  6. H. Jia, J. Yang, X. Li, J. Yang, M. Yang, Y. Liu, and Y. Hao, “Systematic error analysis and compensation for high accuracy star centroid estimation of star tracker,” Sci. China Technol. Sci. 53(11), 3145–3152 (2010).
    [Crossref]
  7. J. Shen, G. Zhang, and X. Wei, “Simulation analysis of dynamic working performance for star trackers,” J. Opt. Soc. Am. A 27(12), 2638–2647 (2010).
    [Crossref] [PubMed]
  8. H. Wang, W. Zhou, X. Cheng, and H. Lin, “Image smearing modeling and verification for strapdown star sensor,” Chin. J. Aeronauti. 25(1), 115–123 (2012).
    [Crossref]
  9. X. Li and H. Zhao, “Analysis of star image centroid accuracy of an APS star sensor in rotation,” Aerospace Control Appl. 35(4), 11–16 (2009).
  10. S. Zhao, Y. Wang, H. Wang, and C. Ji, “Study on dynamic angle-measurement compensation for star sensor,” in Natural Computation (ICNC),2011Seventh International Conference on, (IEEE, 2011), 255–258.
    [Crossref]
  11. Z. Jun, H. Yuncai, W. Li, and L. Da, “Studies on dynamic motion compensation and positioning accuracy on star tracker,” Appl. Opt. 54(28), 8417–8424 (2015).
    [Crossref] [PubMed]
  12. X. Wu and X. Wang, “Multiple blur of star image and the restoration under dynamic conditions,” Acta Astronaut. 68(11–12), 1903–1913 (2011).
  13. T. Sun, F. Xing, Z. You, X. Wang, and B. Li, “Smearing model and restoration of star image under conditions of variable angular velocity and long exposure time,” Opt. Express 22(5), 6009–6024 (2014).
    [Crossref] [PubMed]
  14. W. Zhang, W. Quan, and L. Guo, “Blurred star image processing for star sensors under dynamic conditions,” Sensors (Basel) 12(12), 6712–6726 (2012).
    [Crossref] [PubMed]
  15. C. S. Liu, L. H. Hu, G. B. Liu, B. Yang, and A. J. Li, “Kinematic model for the space-variant image motion of star sensors under dynamical conditions,” Opt. Eng. 54(6), 063104 (2015).
    [Crossref]
  16. W. Hou, H. Liu, Z. Lei, Q. Yu, X. Liu, and J. Dong, “Smeared star spot location estimation using directional integral method,” Appl. Opt. 53(10), 2073–2086 (2014).
    [Crossref] [PubMed]
  17. C. A. Nardell, J. Wertz, and P. B. Hays, “Image processing, simulation and performance predictions for the MicroMak star tracker,” in Optics & Photonics 2005, (International Society for Optics and Photonics, 2005), 59160U–59160U–59112.
  18. T. Dzamba and J. Enright, “Optical trades for evolving a small arcsecond star tracker,” in Aerospace Conference,2013IEEE, (IEEE, 2013), pp. 1–9.
    [Crossref]
  19. T. Dzamba and J. Enright, “Ground testing strategies for verifying the slew rate tolerance of star trackers,” Sensors (Basel) 14(3), 3939–3964 (2014).
    [Crossref] [PubMed]
  20. X. Wei, J. Xu, J. Li, J. Yan, and G. Zhang, “S-curve centroiding error correction for star sensor,” Acta Astronaut. 99, 231–241 (2014).
    [Crossref]
  21. J. Yang, B. Liang, T. Zhang, and J. Song, “A novel systematic error compensation algorithm based on least squares support vector regression for star sensor image centroid estimation,” Sensors (Basel) 11(12), 7341–7363 (2011).
    [Crossref] [PubMed]
  22. X. Wei, W. Tan, J. Li, and G. Zhang, “Exposure Time Optimization for Highly Dynamic Star Trackers,” Sensors (Basel) 14(3), 4914–4931 (2014).
    [Crossref] [PubMed]
  23. B. Shen, J. Tan, J. Yang, and J. Liao, “Exposure Time Optimization of the Star Sensor,” Opto-Electron. Eng. 12, 008 (2009).
  24. E. Aretskin-Hariton and A. J. Swank, “Star Tracker Performance Estimate with IMU,” in AIAA Guidance, Navigation, and Control Conference (American Institute of Aeronautics and Astronautics, 2015), pp 0334.

2015 (2)

Z. Jun, H. Yuncai, W. Li, and L. Da, “Studies on dynamic motion compensation and positioning accuracy on star tracker,” Appl. Opt. 54(28), 8417–8424 (2015).
[Crossref] [PubMed]

C. S. Liu, L. H. Hu, G. B. Liu, B. Yang, and A. J. Li, “Kinematic model for the space-variant image motion of star sensors under dynamical conditions,” Opt. Eng. 54(6), 063104 (2015).
[Crossref]

2014 (5)

W. Hou, H. Liu, Z. Lei, Q. Yu, X. Liu, and J. Dong, “Smeared star spot location estimation using directional integral method,” Appl. Opt. 53(10), 2073–2086 (2014).
[Crossref] [PubMed]

T. Dzamba and J. Enright, “Ground testing strategies for verifying the slew rate tolerance of star trackers,” Sensors (Basel) 14(3), 3939–3964 (2014).
[Crossref] [PubMed]

X. Wei, J. Xu, J. Li, J. Yan, and G. Zhang, “S-curve centroiding error correction for star sensor,” Acta Astronaut. 99, 231–241 (2014).
[Crossref]

T. Sun, F. Xing, Z. You, X. Wang, and B. Li, “Smearing model and restoration of star image under conditions of variable angular velocity and long exposure time,” Opt. Express 22(5), 6009–6024 (2014).
[Crossref] [PubMed]

X. Wei, W. Tan, J. Li, and G. Zhang, “Exposure Time Optimization for Highly Dynamic Star Trackers,” Sensors (Basel) 14(3), 4914–4931 (2014).
[Crossref] [PubMed]

2012 (2)

W. Zhang, W. Quan, and L. Guo, “Blurred star image processing for star sensors under dynamic conditions,” Sensors (Basel) 12(12), 6712–6726 (2012).
[Crossref] [PubMed]

H. Wang, W. Zhou, X. Cheng, and H. Lin, “Image smearing modeling and verification for strapdown star sensor,” Chin. J. Aeronauti. 25(1), 115–123 (2012).
[Crossref]

2011 (2)

X. Wu and X. Wang, “Multiple blur of star image and the restoration under dynamic conditions,” Acta Astronaut. 68(11–12), 1903–1913 (2011).

J. Yang, B. Liang, T. Zhang, and J. Song, “A novel systematic error compensation algorithm based on least squares support vector regression for star sensor image centroid estimation,” Sensors (Basel) 11(12), 7341–7363 (2011).
[Crossref] [PubMed]

2010 (2)

H. Jia, J. Yang, X. Li, J. Yang, M. Yang, Y. Liu, and Y. Hao, “Systematic error analysis and compensation for high accuracy star centroid estimation of star tracker,” Sci. China Technol. Sci. 53(11), 3145–3152 (2010).
[Crossref]

J. Shen, G. Zhang, and X. Wei, “Simulation analysis of dynamic working performance for star trackers,” J. Opt. Soc. Am. A 27(12), 2638–2647 (2010).
[Crossref] [PubMed]

2009 (2)

X. Li and H. Zhao, “Analysis of star image centroid accuracy of an APS star sensor in rotation,” Aerospace Control Appl. 35(4), 11–16 (2009).

B. Shen, J. Tan, J. Yang, and J. Liao, “Exposure Time Optimization of the Star Sensor,” Opto-Electron. Eng. 12, 008 (2009).

2004 (1)

C. C. Liebe, K. Gromov, and D. M. Meller, “Toward a stellar gyroscope for spacecraft attitude determination,” J. Guid. Control Dyn. 27(1), 91–99 (2004).
[Crossref]

2003 (1)

G. Rufino and D. Accardo, “Enhancement of the centroiding algorithm for star tracker measure refinement,” Acta Astronaut. 53(2), 135–147 (2003).
[Crossref]

2002 (2)

C. C. Liebe, “Accuracy performance of star trackers-a tutorial,” IEEE Trans. Aerosp. Electron. Syst. 38(2), 587–599 (2002).
[Crossref]

M. A. Samaan, T. C. Pollock, and J. L. Junkins, “Predictive centroiding for star trackers with the effect of image smear,” J. Astronaut. Sci. 50(1), 113–123 (2002).

Accardo, D.

G. Rufino and D. Accardo, “Enhancement of the centroiding algorithm for star tracker measure refinement,” Acta Astronaut. 53(2), 135–147 (2003).
[Crossref]

Cheng, X.

H. Wang, W. Zhou, X. Cheng, and H. Lin, “Image smearing modeling and verification for strapdown star sensor,” Chin. J. Aeronauti. 25(1), 115–123 (2012).
[Crossref]

Cunningham, T. J.

B. R. Hancock, R. C. Stirbl, T. J. Cunningham, B. Pain, C. J. Wrigley, and P. G. Ringold, “CMOS active pixel sensor specific performance effects on star tracker/imager position accuracy”, in Symposium on Integrated Optics, (International Society for Optics and Photonics, 2001), pp. 43–53.
[Crossref]

Da, L.

Dong, J.

Dzamba, T.

T. Dzamba and J. Enright, “Ground testing strategies for verifying the slew rate tolerance of star trackers,” Sensors (Basel) 14(3), 3939–3964 (2014).
[Crossref] [PubMed]

Enright, J.

T. Dzamba and J. Enright, “Ground testing strategies for verifying the slew rate tolerance of star trackers,” Sensors (Basel) 14(3), 3939–3964 (2014).
[Crossref] [PubMed]

Gromov, K.

C. C. Liebe, K. Gromov, and D. M. Meller, “Toward a stellar gyroscope for spacecraft attitude determination,” J. Guid. Control Dyn. 27(1), 91–99 (2004).
[Crossref]

Guo, L.

W. Zhang, W. Quan, and L. Guo, “Blurred star image processing for star sensors under dynamic conditions,” Sensors (Basel) 12(12), 6712–6726 (2012).
[Crossref] [PubMed]

Hancock, B. R.

B. R. Hancock, R. C. Stirbl, T. J. Cunningham, B. Pain, C. J. Wrigley, and P. G. Ringold, “CMOS active pixel sensor specific performance effects on star tracker/imager position accuracy”, in Symposium on Integrated Optics, (International Society for Optics and Photonics, 2001), pp. 43–53.
[Crossref]

Hao, Y.

H. Jia, J. Yang, X. Li, J. Yang, M. Yang, Y. Liu, and Y. Hao, “Systematic error analysis and compensation for high accuracy star centroid estimation of star tracker,” Sci. China Technol. Sci. 53(11), 3145–3152 (2010).
[Crossref]

Hou, W.

Hu, L. H.

C. S. Liu, L. H. Hu, G. B. Liu, B. Yang, and A. J. Li, “Kinematic model for the space-variant image motion of star sensors under dynamical conditions,” Opt. Eng. 54(6), 063104 (2015).
[Crossref]

Jia, H.

H. Jia, J. Yang, X. Li, J. Yang, M. Yang, Y. Liu, and Y. Hao, “Systematic error analysis and compensation for high accuracy star centroid estimation of star tracker,” Sci. China Technol. Sci. 53(11), 3145–3152 (2010).
[Crossref]

Jun, Z.

Junkins, J. L.

M. A. Samaan, T. C. Pollock, and J. L. Junkins, “Predictive centroiding for star trackers with the effect of image smear,” J. Astronaut. Sci. 50(1), 113–123 (2002).

Lei, Z.

Li, A. J.

C. S. Liu, L. H. Hu, G. B. Liu, B. Yang, and A. J. Li, “Kinematic model for the space-variant image motion of star sensors under dynamical conditions,” Opt. Eng. 54(6), 063104 (2015).
[Crossref]

Li, B.

Li, J.

X. Wei, J. Xu, J. Li, J. Yan, and G. Zhang, “S-curve centroiding error correction for star sensor,” Acta Astronaut. 99, 231–241 (2014).
[Crossref]

X. Wei, W. Tan, J. Li, and G. Zhang, “Exposure Time Optimization for Highly Dynamic Star Trackers,” Sensors (Basel) 14(3), 4914–4931 (2014).
[Crossref] [PubMed]

Li, W.

Li, X.

H. Jia, J. Yang, X. Li, J. Yang, M. Yang, Y. Liu, and Y. Hao, “Systematic error analysis and compensation for high accuracy star centroid estimation of star tracker,” Sci. China Technol. Sci. 53(11), 3145–3152 (2010).
[Crossref]

X. Li and H. Zhao, “Analysis of star image centroid accuracy of an APS star sensor in rotation,” Aerospace Control Appl. 35(4), 11–16 (2009).

Liang, B.

J. Yang, B. Liang, T. Zhang, and J. Song, “A novel systematic error compensation algorithm based on least squares support vector regression for star sensor image centroid estimation,” Sensors (Basel) 11(12), 7341–7363 (2011).
[Crossref] [PubMed]

Liao, J.

B. Shen, J. Tan, J. Yang, and J. Liao, “Exposure Time Optimization of the Star Sensor,” Opto-Electron. Eng. 12, 008 (2009).

Liebe, C. C.

C. C. Liebe, K. Gromov, and D. M. Meller, “Toward a stellar gyroscope for spacecraft attitude determination,” J. Guid. Control Dyn. 27(1), 91–99 (2004).
[Crossref]

C. C. Liebe, “Accuracy performance of star trackers-a tutorial,” IEEE Trans. Aerosp. Electron. Syst. 38(2), 587–599 (2002).
[Crossref]

Lin, H.

H. Wang, W. Zhou, X. Cheng, and H. Lin, “Image smearing modeling and verification for strapdown star sensor,” Chin. J. Aeronauti. 25(1), 115–123 (2012).
[Crossref]

Liu, C. S.

C. S. Liu, L. H. Hu, G. B. Liu, B. Yang, and A. J. Li, “Kinematic model for the space-variant image motion of star sensors under dynamical conditions,” Opt. Eng. 54(6), 063104 (2015).
[Crossref]

Liu, G. B.

C. S. Liu, L. H. Hu, G. B. Liu, B. Yang, and A. J. Li, “Kinematic model for the space-variant image motion of star sensors under dynamical conditions,” Opt. Eng. 54(6), 063104 (2015).
[Crossref]

Liu, H.

Liu, X.

Liu, Y.

H. Jia, J. Yang, X. Li, J. Yang, M. Yang, Y. Liu, and Y. Hao, “Systematic error analysis and compensation for high accuracy star centroid estimation of star tracker,” Sci. China Technol. Sci. 53(11), 3145–3152 (2010).
[Crossref]

Meller, D. M.

C. C. Liebe, K. Gromov, and D. M. Meller, “Toward a stellar gyroscope for spacecraft attitude determination,” J. Guid. Control Dyn. 27(1), 91–99 (2004).
[Crossref]

Pain, B.

B. R. Hancock, R. C. Stirbl, T. J. Cunningham, B. Pain, C. J. Wrigley, and P. G. Ringold, “CMOS active pixel sensor specific performance effects on star tracker/imager position accuracy”, in Symposium on Integrated Optics, (International Society for Optics and Photonics, 2001), pp. 43–53.
[Crossref]

Pollock, T. C.

M. A. Samaan, T. C. Pollock, and J. L. Junkins, “Predictive centroiding for star trackers with the effect of image smear,” J. Astronaut. Sci. 50(1), 113–123 (2002).

Quan, W.

W. Zhang, W. Quan, and L. Guo, “Blurred star image processing for star sensors under dynamic conditions,” Sensors (Basel) 12(12), 6712–6726 (2012).
[Crossref] [PubMed]

Ringold, P. G.

B. R. Hancock, R. C. Stirbl, T. J. Cunningham, B. Pain, C. J. Wrigley, and P. G. Ringold, “CMOS active pixel sensor specific performance effects on star tracker/imager position accuracy”, in Symposium on Integrated Optics, (International Society for Optics and Photonics, 2001), pp. 43–53.
[Crossref]

Rufino, G.

G. Rufino and D. Accardo, “Enhancement of the centroiding algorithm for star tracker measure refinement,” Acta Astronaut. 53(2), 135–147 (2003).
[Crossref]

Samaan, M. A.

M. A. Samaan, T. C. Pollock, and J. L. Junkins, “Predictive centroiding for star trackers with the effect of image smear,” J. Astronaut. Sci. 50(1), 113–123 (2002).

Shen, B.

B. Shen, J. Tan, J. Yang, and J. Liao, “Exposure Time Optimization of the Star Sensor,” Opto-Electron. Eng. 12, 008 (2009).

Shen, J.

Song, J.

J. Yang, B. Liang, T. Zhang, and J. Song, “A novel systematic error compensation algorithm based on least squares support vector regression for star sensor image centroid estimation,” Sensors (Basel) 11(12), 7341–7363 (2011).
[Crossref] [PubMed]

Stirbl, R. C.

B. R. Hancock, R. C. Stirbl, T. J. Cunningham, B. Pain, C. J. Wrigley, and P. G. Ringold, “CMOS active pixel sensor specific performance effects on star tracker/imager position accuracy”, in Symposium on Integrated Optics, (International Society for Optics and Photonics, 2001), pp. 43–53.
[Crossref]

Sun, T.

Tan, J.

B. Shen, J. Tan, J. Yang, and J. Liao, “Exposure Time Optimization of the Star Sensor,” Opto-Electron. Eng. 12, 008 (2009).

Tan, W.

X. Wei, W. Tan, J. Li, and G. Zhang, “Exposure Time Optimization for Highly Dynamic Star Trackers,” Sensors (Basel) 14(3), 4914–4931 (2014).
[Crossref] [PubMed]

Wang, H.

H. Wang, W. Zhou, X. Cheng, and H. Lin, “Image smearing modeling and verification for strapdown star sensor,” Chin. J. Aeronauti. 25(1), 115–123 (2012).
[Crossref]

Wang, X.

T. Sun, F. Xing, Z. You, X. Wang, and B. Li, “Smearing model and restoration of star image under conditions of variable angular velocity and long exposure time,” Opt. Express 22(5), 6009–6024 (2014).
[Crossref] [PubMed]

X. Wu and X. Wang, “Multiple blur of star image and the restoration under dynamic conditions,” Acta Astronaut. 68(11–12), 1903–1913 (2011).

Wei, X.

X. Wei, W. Tan, J. Li, and G. Zhang, “Exposure Time Optimization for Highly Dynamic Star Trackers,” Sensors (Basel) 14(3), 4914–4931 (2014).
[Crossref] [PubMed]

X. Wei, J. Xu, J. Li, J. Yan, and G. Zhang, “S-curve centroiding error correction for star sensor,” Acta Astronaut. 99, 231–241 (2014).
[Crossref]

J. Shen, G. Zhang, and X. Wei, “Simulation analysis of dynamic working performance for star trackers,” J. Opt. Soc. Am. A 27(12), 2638–2647 (2010).
[Crossref] [PubMed]

Wrigley, C. J.

B. R. Hancock, R. C. Stirbl, T. J. Cunningham, B. Pain, C. J. Wrigley, and P. G. Ringold, “CMOS active pixel sensor specific performance effects on star tracker/imager position accuracy”, in Symposium on Integrated Optics, (International Society for Optics and Photonics, 2001), pp. 43–53.
[Crossref]

Wu, X.

X. Wu and X. Wang, “Multiple blur of star image and the restoration under dynamic conditions,” Acta Astronaut. 68(11–12), 1903–1913 (2011).

Xing, F.

Xu, J.

X. Wei, J. Xu, J. Li, J. Yan, and G. Zhang, “S-curve centroiding error correction for star sensor,” Acta Astronaut. 99, 231–241 (2014).
[Crossref]

Yan, J.

X. Wei, J. Xu, J. Li, J. Yan, and G. Zhang, “S-curve centroiding error correction for star sensor,” Acta Astronaut. 99, 231–241 (2014).
[Crossref]

Yang, B.

C. S. Liu, L. H. Hu, G. B. Liu, B. Yang, and A. J. Li, “Kinematic model for the space-variant image motion of star sensors under dynamical conditions,” Opt. Eng. 54(6), 063104 (2015).
[Crossref]

Yang, J.

J. Yang, B. Liang, T. Zhang, and J. Song, “A novel systematic error compensation algorithm based on least squares support vector regression for star sensor image centroid estimation,” Sensors (Basel) 11(12), 7341–7363 (2011).
[Crossref] [PubMed]

H. Jia, J. Yang, X. Li, J. Yang, M. Yang, Y. Liu, and Y. Hao, “Systematic error analysis and compensation for high accuracy star centroid estimation of star tracker,” Sci. China Technol. Sci. 53(11), 3145–3152 (2010).
[Crossref]

H. Jia, J. Yang, X. Li, J. Yang, M. Yang, Y. Liu, and Y. Hao, “Systematic error analysis and compensation for high accuracy star centroid estimation of star tracker,” Sci. China Technol. Sci. 53(11), 3145–3152 (2010).
[Crossref]

B. Shen, J. Tan, J. Yang, and J. Liao, “Exposure Time Optimization of the Star Sensor,” Opto-Electron. Eng. 12, 008 (2009).

Yang, M.

H. Jia, J. Yang, X. Li, J. Yang, M. Yang, Y. Liu, and Y. Hao, “Systematic error analysis and compensation for high accuracy star centroid estimation of star tracker,” Sci. China Technol. Sci. 53(11), 3145–3152 (2010).
[Crossref]

You, Z.

Yu, Q.

Yuncai, H.

Zhang, G.

X. Wei, W. Tan, J. Li, and G. Zhang, “Exposure Time Optimization for Highly Dynamic Star Trackers,” Sensors (Basel) 14(3), 4914–4931 (2014).
[Crossref] [PubMed]

X. Wei, J. Xu, J. Li, J. Yan, and G. Zhang, “S-curve centroiding error correction for star sensor,” Acta Astronaut. 99, 231–241 (2014).
[Crossref]

J. Shen, G. Zhang, and X. Wei, “Simulation analysis of dynamic working performance for star trackers,” J. Opt. Soc. Am. A 27(12), 2638–2647 (2010).
[Crossref] [PubMed]

Zhang, T.

J. Yang, B. Liang, T. Zhang, and J. Song, “A novel systematic error compensation algorithm based on least squares support vector regression for star sensor image centroid estimation,” Sensors (Basel) 11(12), 7341–7363 (2011).
[Crossref] [PubMed]

Zhang, W.

W. Zhang, W. Quan, and L. Guo, “Blurred star image processing for star sensors under dynamic conditions,” Sensors (Basel) 12(12), 6712–6726 (2012).
[Crossref] [PubMed]

Zhao, H.

X. Li and H. Zhao, “Analysis of star image centroid accuracy of an APS star sensor in rotation,” Aerospace Control Appl. 35(4), 11–16 (2009).

Zhou, W.

H. Wang, W. Zhou, X. Cheng, and H. Lin, “Image smearing modeling and verification for strapdown star sensor,” Chin. J. Aeronauti. 25(1), 115–123 (2012).
[Crossref]

Acta Astronaut. (3)

G. Rufino and D. Accardo, “Enhancement of the centroiding algorithm for star tracker measure refinement,” Acta Astronaut. 53(2), 135–147 (2003).
[Crossref]

X. Wu and X. Wang, “Multiple blur of star image and the restoration under dynamic conditions,” Acta Astronaut. 68(11–12), 1903–1913 (2011).

X. Wei, J. Xu, J. Li, J. Yan, and G. Zhang, “S-curve centroiding error correction for star sensor,” Acta Astronaut. 99, 231–241 (2014).
[Crossref]

Aerospace Control Appl. (1)

X. Li and H. Zhao, “Analysis of star image centroid accuracy of an APS star sensor in rotation,” Aerospace Control Appl. 35(4), 11–16 (2009).

Appl. Opt. (2)

Chin. J. Aeronauti. (1)

H. Wang, W. Zhou, X. Cheng, and H. Lin, “Image smearing modeling and verification for strapdown star sensor,” Chin. J. Aeronauti. 25(1), 115–123 (2012).
[Crossref]

IEEE Trans. Aerosp. Electron. Syst. (1)

C. C. Liebe, “Accuracy performance of star trackers-a tutorial,” IEEE Trans. Aerosp. Electron. Syst. 38(2), 587–599 (2002).
[Crossref]

J. Astronaut. Sci. (1)

M. A. Samaan, T. C. Pollock, and J. L. Junkins, “Predictive centroiding for star trackers with the effect of image smear,” J. Astronaut. Sci. 50(1), 113–123 (2002).

J. Guid. Control Dyn. (1)

C. C. Liebe, K. Gromov, and D. M. Meller, “Toward a stellar gyroscope for spacecraft attitude determination,” J. Guid. Control Dyn. 27(1), 91–99 (2004).
[Crossref]

J. Opt. Soc. Am. A (1)

Opt. Eng. (1)

C. S. Liu, L. H. Hu, G. B. Liu, B. Yang, and A. J. Li, “Kinematic model for the space-variant image motion of star sensors under dynamical conditions,” Opt. Eng. 54(6), 063104 (2015).
[Crossref]

Opt. Express (1)

Opto-Electron. Eng. (1)

B. Shen, J. Tan, J. Yang, and J. Liao, “Exposure Time Optimization of the Star Sensor,” Opto-Electron. Eng. 12, 008 (2009).

Sci. China Technol. Sci. (1)

H. Jia, J. Yang, X. Li, J. Yang, M. Yang, Y. Liu, and Y. Hao, “Systematic error analysis and compensation for high accuracy star centroid estimation of star tracker,” Sci. China Technol. Sci. 53(11), 3145–3152 (2010).
[Crossref]

Sensors (Basel) (4)

W. Zhang, W. Quan, and L. Guo, “Blurred star image processing for star sensors under dynamic conditions,” Sensors (Basel) 12(12), 6712–6726 (2012).
[Crossref] [PubMed]

T. Dzamba and J. Enright, “Ground testing strategies for verifying the slew rate tolerance of star trackers,” Sensors (Basel) 14(3), 3939–3964 (2014).
[Crossref] [PubMed]

J. Yang, B. Liang, T. Zhang, and J. Song, “A novel systematic error compensation algorithm based on least squares support vector regression for star sensor image centroid estimation,” Sensors (Basel) 11(12), 7341–7363 (2011).
[Crossref] [PubMed]

X. Wei, W. Tan, J. Li, and G. Zhang, “Exposure Time Optimization for Highly Dynamic Star Trackers,” Sensors (Basel) 14(3), 4914–4931 (2014).
[Crossref] [PubMed]

Other (5)

E. Aretskin-Hariton and A. J. Swank, “Star Tracker Performance Estimate with IMU,” in AIAA Guidance, Navigation, and Control Conference (American Institute of Aeronautics and Astronautics, 2015), pp 0334.

C. A. Nardell, J. Wertz, and P. B. Hays, “Image processing, simulation and performance predictions for the MicroMak star tracker,” in Optics & Photonics 2005, (International Society for Optics and Photonics, 2005), 59160U–59160U–59112.

T. Dzamba and J. Enright, “Optical trades for evolving a small arcsecond star tracker,” in Aerospace Conference,2013IEEE, (IEEE, 2013), pp. 1–9.
[Crossref]

S. Zhao, Y. Wang, H. Wang, and C. Ji, “Study on dynamic angle-measurement compensation for star sensor,” in Natural Computation (ICNC),2011Seventh International Conference on, (IEEE, 2011), 255–258.
[Crossref]

B. R. Hancock, R. C. Stirbl, T. J. Cunningham, B. Pain, C. J. Wrigley, and P. G. Ringold, “CMOS active pixel sensor specific performance effects on star tracker/imager position accuracy”, in Symposium on Integrated Optics, (International Society for Optics and Photonics, 2001), pp. 43–53.
[Crossref]

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Figures (16)

Fig. 1
Fig. 1 Dynamic imaging process of a star spot.
Fig. 2
Fig. 2 Rotation transformation of the star streak.
Fig. 3
Fig. 3 Discrete star streak image.
Fig. 4
Fig. 4 Two types of rotation. (a) Rotation about boresight; (b) Rotation about cross-boresight axis.
Fig. 5
Fig. 5 Centroiding error of the non-linear moion. (a) Boresight rotation; (b) Cross-boresight rotation.
Fig. 6
Fig. 6 Centroiding error εxy vs. smear length.
Fig. 7
Fig. 7 σx,sys vs. Gaussian radius ρ and smear length L.
Fig. 8
Fig. 8 Total systematic error σ sys vs. Gaussian radius ρ and smear length L.
Fig. 9
Fig. 9 Total random error vs. smear length under different Gaussian radii.
Fig. 10
Fig. 10 random error of centroiding vs. smear length.
Fig. 11
Fig. 11 Optimal exposure time vs. angular velocity under different stellar magnitudes.
Fig. 12
Fig. 12 Centroiding error vs. Gaussian radius.
Fig. 13
Fig. 13 Comparison of centroiding errors of different methods.
Fig. 14
Fig. 14 Centroiding error vs. smear length L.
Fig. 15
Fig. 15 Setup of the night sky experiment.
Fig. 16
Fig. 16 Centroiding error vs. smear length under different velocities in the night sky experiment.

Tables (1)

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Table 1 Parameters of the CMV4000 Image Sensor

Equations (53)

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f PSF ( x ) = 1 2 π ρ exp ( x 2 2 ρ 2 ) ,
f LSSF ( x ) = 1 L x L x f PSF ( u ) d u = 1 2 L [ erf ( x 2 ρ ) erf ( x L 2 ρ ) ] ,
E sta ( x , y ) = Φ T 2 π ρ 2 exp [ ( x x c ) 2 + ( y y c ) 2 2 ρ 2 ] = Φ T f PSF ( x x c ) f PSF ( y y c ) ,
E dyn ( x , y ) = 0 T Φ 2 π ρ 2 exp { [ x x c ( t ) ] 2 + [ y y c ( t ) ] 2 2 ρ 2 } d t = Φ 0 T f PSF ( x x c ( t ) ) f PSF ( y y c ( t ) ) d t .
E dyn ( x , y ) = Φ 0 T f PSF ( x x 0 v t ) f PSF ( y y 0 ) d t = Φ 2 v [ erf ( x x 0 2 ρ ) erf ( x x 0 v T 2 ρ ) ] f PSF ( y y 0 ) = Φ T f LSSF ( x x 0 ) f PSF ( y y 0 ) .
E dyn ( x , y ) = Φ 0 T f PSF ( x x 0 v t cos θ ) f PSF ( y y 0 v t sin θ ) d t = Φ 2 v [ erf ( u 2 ρ ) erf ( u v T 2 ρ ) ] 1 2 π ρ exp ( v 2 2 ρ 2 ) = Φ T f LSSF ( u ) f PSF ( v ) ,
{ x = X cos θ Y sin θ y = X sin θ + Y cos θ .
E dyn ( X , Y ) = Φ T f LSSF ( X X 0 ) f PSF ( Y Y 0 ) .
E max ( X , Y ) = Φ T f LSSF ( v T / 2 ) f PSF ( y 0 ) = Φ 2 π ρ v erf ( v T 2 2 ρ ) .
E lim = lim T Φ 2 π ρ v erf ( v T 2 2 ρ ) = Φ 2 π ρ v .
I i j = y j 0.5 y j + 0.5 x i 0.5 x i + 0.5 η QE K E dyn ( x , y ) d x d y = Φ T η QE K x i 0.5 x i + 0.5 f LSSF ( x x 0 ) d x y j 0.5 y j + 0.5 f PSF ( y y 0 ) d y , = Φ T η QE K F LSSF ( x i x 0 ) F PSF ( y j y 0 )
F LSSF ( x i ) = x i 0.5 x i + 0.5 f LSSF ( x ) d x = 1 2 L { ( x i + 0.5 ) erf [ ( x i + 0.5 ) / ( 2 ρ ) ] + 2 ρ 2 f PSF ( x i + 0.5 ) ( x i 0.5 ) erf [ ( x i 0.5 ) / ( 2 ρ ) ] 2 ρ 2 f PSF ( x i 0.5 ) ( x i + 0.5 L ) erf [ ( x i + 0.5 L ) / ( 2 ρ ) ] 2 ρ 2 f PSF ( x i + 0.5 L ) ( x i 0.5 L ) erf [ ( x i 0.5 L ) / ( 2 ρ ) ] + 2 ρ 2 f PSF ( x i 0.5 L ) } .
I i = j I i j = Φ T η QE K F LSSF ( x i x 0 ) .
I total = i j I i j = i I i = Φ T η QE K .
L z ( t ) = f tan θ Δ θ z = f tan θ ω z t = v z t ,
L x y ( t ) = f [ tan ( θ + Δ θ x y ) tan θ ] = f [ tan ( θ + ω x y t ) tan θ ] .
v x y ( t ) = d L x y ( t ) d t = f ω x y [ 1 + tan 2 ( θ + ω x y t ) ] f ω x y .
ε z = | P ' P | = | O P | | O P ' | = r ( 1 cos ω z t 2 ) .
ε z = r ( 1 cos ω z t 2 ) h 2 ( 1 cos ω z t 2 ) = 4.76 × 10 6 h .
ε z 4.76 × 10 6 h / d = 0.0097 pixels,
| A P | = 1 T 0 T L x y ( t ) d t = f ω x y T ln cos θ cos ( θ + ω x y T ) f tan θ ,
| A P ' | = v x y ( 0 ) T 2 = f ω x y [ 1 + tan 2 ( θ ) ] T 2 .
ε x y = | A P | | A P ' | = f ω x y T ln cos θ cos ( θ + ω x y T ) f tan θ f ω x y [ 1 + tan 2 ( θ ) ] T 2 f ω x y T ln cos ( θ FOV / 2 ) cos ( θ FOV / 2 + ω x y T ) f tan θ FOV 2 f ω x y [ 1 + tan 2 θ FOV 2 ] T 2
x ¯ = x E dyn ( x , y ) d x d y E dyn ( x , y ) d x d y , y ¯ = y E dyn ( x , y ) d x d y E dyn ( x , y ) d x d y .
x ¯ = x f LSSF ( x x 0 ) d x = x 0 + v T / 2 = x c . y ¯ = y f PSF ( y y 0 ) d y = y 0 = y c
x ¯ = i j x i I i j i j I i j ,
x ¯ = i x i I i i I i .
δ x = x ¯ x c = i x i I i i I i ( x 0 + L 2 ) .
δ x = i x i Φ T η QE K F LSSF ( x i x 0 ) Φ T η QE K ( x 0 + L 2 ) = 1 2 L i x i x i 0.5 x i + 0.5 [ erf ( x x 0 2 ρ ) erf ( x x 0 L 2 ρ ) ] d x ( x 0 + L 2 ) .
σ x , sys ( σ , L ) = [ 0.5 0.5 δ x 2 ( ρ , L , x 0 ) d x 0 ] 1 / 2 .
l = 2 × 3 ρ 0.5 + 1 + L , w = 2 × 3 ρ 0.5 + 1 ,
σ y , sys ( ρ ) = lim L 0 σ x , sys ( ρ , L ) .
σ sys = σ x , sys 2 + σ y , sys 2 .
σ x , ran 2 = i ( x ¯ x i ) 2 n x , i 2 + i ( x ¯ I i ) 2 n I , i 2 = i ( I i I toltal ) 2 n x , i 2 + i ( x i x ¯ I toltal ) 2 n I , i 2 ,
σ x , ran 2 = i ( x i x ¯ I toltal ) 2 n I , i 2 .
n add 2 = n dc 2 + n read 2 + n ADC 2 = I dark T + n read 2 + 1 / ( 12 K 2 ) .
n I , i j 2 = K 2 ( n i j , shot 2 + n add 2 ) = K I i j + K 2 n add 2 .
n I , i 2 = j n I , i j 2 = K I i + w K 2 n add 2 .
σ x , ran 2 = K I toltal 2 i ( x i x ¯ ) 2 I i + w K 2 n add 2 I toltal 2 i ( x i x ¯ ) 2 = 1 Φ T η QE 3 ρ + x 0 x 0 + L + 3 ρ ( x i x 0 L / 2 ) 2 f LSSF ( x i ) d x i + w n add 2 ( Φ T η QE ) 2 3 ρ + x 0 x 0 + L + 3 ρ ( x i x 0 L / 2 ) 2 d x i .
σ x , shot 2 = 1 Φ T η QE 3 ρ + x 0 x 0 + L + 3 ρ ( x i x 0 L / 2 ) 2 f LSSF ( x i ) d x i 1 Φ T η QE ( x i L / 2 ) 2 f LSSF ( x i ) d x i = v Φ η QE L ( 1 12 L 2 + ρ 2 ) .
σ x , add 2 = w n add 2 ( Φ T η QE ) 2 3 ρ + x 0 x 0 + L + 3 ρ ( x i x 0 L / 2 ) 2 d x i = 6 ρ v 2 n add 2 Φ 2 η QE 2 L 2 1 12 ( L + 6 ρ ) 3 .
σ y , shot 2 = K I toltal 2 j ( y j y ¯ ) 2 I j = v ρ 2 Φ η QE L .
σ y , add 2 = K 2 n add 2 l I toltal 2 j ( y j y ¯ ) 2 = 18 v 2 n add 2 ( L + 6 ρ ) ρ 3 ( Φ η QE L ) 2 .
σ ran 2 = ( σ x , shot 2 + σ y , shot 2 ) + ( σ x , add 2 + σ y , add 2 ) = v Φ η QE L ( 1 12 L 2 + 2 ρ 2 ) + v 2 n add 2 ρ ( L + 6 ρ ) 2 ( Φ η QE L ) 2 ( L 2 + 12 L ρ + 72 ρ 2 ) .
σ c 2 = σ sys 2 + σ ran 2 .
Φ = τ E 0 / ε 2.512 m π D 2 / 4 ,
v f ω = h / 2 d tan ( θ FOV / 2 ) ω = N pix / 2 tan ( θ FOV / 2 ) ω ,
σ shot 2 = σ x ,shot 2 + σ y ,shot 2 = v Φ η QE L ( 1 12 L 2 + 2 ρ 2 ) .
L 1 = 2 6 ρ 4.899 ρ ,
σ add 2 = σ x ,add 2 + σ y ,add 2 = 18 v 2 n add 2 ρ ( L + 6 ρ ) ( Φ η QE L ) 2 ( L 2 + 12 L ρ + 72 ρ 2 ) .
L 2 = 2 ρ [ ( 54 + 6 33 ) 1 / 3 + 12 ( 54 + 6 33 ) 1 / 3 ] 14.289 ρ .
L o = arg L { min [ σ ran 2 ( L ) ] } , L [ L 1 , L 2 ] .
ρ o = arg ρ { min [ σ c 2 ( ρ ) ] } = arg ρ { min [ σ sys 2 ( ρ ) + σ ran 2 ( ρ ) ] } .

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