Angular velocity estimation from measurement vectors of star tracker Hai-bo Liu,* Jun-cai Yang, Wen-jun Yi, Jiong-qi Wang, Jian-kun Yang, Xiu-jian Li, and Ji-chun Tan National University of Defense Technology, Changsha, Hunan, 410073, China *Corresponding author:
[email protected] Received 11 January 2012; revised 7 April 2012; accepted 7 April 2012; posted 9 April 2012 (Doc. ID 161291); published 1 June 2012
In most spacecraft, there is a need to know the craft’s angular rate. Approaches with least squares and an adaptive Kalman filter are proposed for estimating the angular rate directly from the star tracker measurements. In these approaches, only knowledge of the vector measurements and sampling interval is required. The designed adaptive Kalman filter can filter out noise without information of the dynamic model and inertia dyadic. To verify the proposed estimation approaches, simulations based on the orbit data of the challenging minisatellite payload (CHAMP) satellite and experimental tests with night-sky observation are performed. Both the simulations and experimental testing results have demonstrated that the proposed approach performs well in terms of accuracy, robustness, and performance. © 2012 Optical Society of America OCIS codes: 120.6085, 350.6090.
1. Introduction
The knowledge of a spacecraft’s angular rate is required for the tasks of spacecraft attitude control and attitude determination [1]. Generally, the information about angular rate is obtained from gyros [2]. However, the rate gyros have a tendency to degrade or fail in orbit. In such cases, there is a need to estimate angular rate in the absence of rate gyros. Besides, some small spacecraft designs do not carry gyroscopes but, nevertheless, need to determine their angular velocity for attitude control and attitude propagation purposes. Therefore, the inclination now is to do away with gyros and use other means to determine the angular rate [3]. There are several ways to obtain the angular rate in a gyroless spacecraft. When the attitude is known, one can differentiate the attitude and use the kinematics equations that connect the derivative of the attitude with the satellite angular rate to compute the latter [4–6]. For example, if the attitude quaternion q is known, its derivative q_ can be usually 1559-128X/12/163590-09$15.00/0 © 2012 Optical Society of America 3590
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approximated by a finite-difference approach. Then the angular velocity can be computed directly from the kinematics equations. Another approach is using the angular dynamics in some kind of active filter [7–9]. In that case, the dynamic model continuously tracks the total momentum and inertia dyadic, from which the angular rate can be calculated. As will be shown, the delay can be eliminated when using an active filter. However, the accuracy of the rate estimate is dependent on the quality of the dynamic model, the knowledge of the system, and the component of inertia dyadic. Errors are also introduced in the model by external disturbances [10]. J. L. Crassidis proposed a simple approach to determine the angular velocity that merely depends on the knowledge of the body vector measurements, which are obtained directly from the star tracker [11]. The main advantage of Crassidis’s approach is that there are no needs for information of the star reference vectors or attitude. However, it requires one time step ahead to estimate the angular velocity at the current time, which becomes less significant as the sampling frequency decreases. In this paper, we develop a method to estimate the angular velocity by
measurement vectors of the star tracker in real time, and an adaptive Kalman filter is designed on different guide stars in a same-star image frame without information of the dynamic model and inertia dyadic. Actually, this paper integrates the work presented in [12], where a Kalman filter approach has been implemented to improve the star acquisition procedure. 2. Angular Velocity Estimation with Batch Least Squares
In this section the least-squares approach is used to determine the angular velocity from star tracker measurements alone. A.
Preliminaries
The measurement model of the star tracker can be considered as a pinhole imaging system, as shown in Fig. 1, in which the star’s direction vector is provided in the star tracker reference [12]. As shown in Fig. 1, r and vt represent the cataloged vector in the inertial frame O0 -XnYnZn and the measurement direction vector in the star tracker frame o0 -xyz at the time of t, respectively. In theory, r, vt, and the attitude matrix Mt of the star tracker satisfy [13] vt Mtr; where
and
(1) 2
3 −xt 1 vt p 4 −yt 5 2 x t y2 t f 2 f
(2)
" cos α cos δ # r sin α cos δ : sin δ
(3)
observed star location on the detector plane. α; δ represents the right ascension and declination of the associated guide star on the celestial sphere. The derivative with respect to time t of Eq. (1) is [4] ∂vt ∂Mt r −ωs t×Mtr: ∂t ∂t
(4)
In Eq. (4), the three components of the startracker-referenced angular velocity ωs t≜ ω1 t ω2 tω3 tT , where ωi is the angular velocity component along the i axis of the star tracker frame and i 1, 2, 3 for x, y , z, respectively. ωs t× is the cross-product matrix to ωs t. By substituting Eq. (1) into Eq. (4), the following expression can be achieved: ∂vt −ωs t×vt vt×ωs t: ∂t
(5)
According to Eq. (5), only knowledge of the direction vector measurements and their derivative with respect to time t are required to derive an angular velocity estimate. Therefore, the attitude and star reference vectors are not required to be known. Hence, stars do not need to be identified to determine the angular velocity ωs t. B. Least-Squares Estimation Approach
In Eq. (2) and Eq. (3), f represents the focal length of the star tracker camera and x; y represents the
J. L. Crassidis proposed a simple approach to determine the angular velocity that depends only on the knowledge of the body vector measurements [11]. This section repeats the derivation done by Crassidis. However, we redefine the Crassidis’s solution using a backward difference instead of a forward difference. Taking the first-order forward difference approximation, ∂vt∕∂t in Eq. (5) can be obtained as ∂vt vt Δt − vt ≈ ; ∂t Δt
(6)
where Δt is the sampling interval of the star tracker. Substituting Eq. (6) into Eq. (5) yields vt Δt − vt vt×ωs t: Δt
(7)
Assuming that the unit vector vt is used in standard form with σ 2 I 3×3 , where I 3×3 is a 3 × 3 identity matrix, as the measurement noise covariance, Eq. (8) can be cast into a linear least-squares form for all measurement vectors, which leads to [11] n −1 1 X −2 T ^ s t σ vtk × vtk × ω Δt k1 k × Fig. 1. Measurement model of star tracker.
n X
T σ −2 k vtk × vt Δtk ;
(8)
k1
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^ s t is the estimate of ωs t, σ 2k ≡ 2σ 2k ∕Δt2 , and where ω σ k is the standard deviation of the measurement error involving the unit-vector observations. n is the number of observations. The predicted error covariance (i.e., the best available estimate of the error covariance) is approximately given by ^ s t − ωs tω^ s t − ωs tT g Pt Efω X −1 n −2 T σ k vtk × vtk × ;
(9)
k1
where Ef g denotes expectation. Equations (8) and (9) are the angular velocity estimate and the predicted error covariance under the first-order forward difference approximation in Crassidis’s approach. Obviously, knowledge of the same-star measurement vk at the times of t and t Δt is required to estimate ωs t at the time of t. That is to say, there is an estimation delay as it requires one time step ahead to estimate the angular velocity at the current time. Actually, using the backward difference instead of the forward difference, ωs t at the time of t can be estimated with the knowledge of the same-star measurement vk at the times of t − Δt and t but not at t and t Δt. Therefore, the estimation delay can be avoided. Taking the first-order backward difference approximation, Eq. (10) can be obtained: ∂vt vt − vt − Δt ≈ : ∂t Δt
(10)
By replacing Eq. (6) by Eq. (10), Eq. (11) is achieved: vt − vt − Δt vt×ωs t: Δt
(11)
According to Crassidis’s research, the angular velocity estimate can be given as ^ s t − ω ×
n −1 1 X T σ −2 vt × vt × k k Δt k1 k n X
T σ −2 k vtk × vt − Δtk :
(12)
was developed to solve the state estimation problems of a known linear system based on certain assumptions. It is one of the powerful tools that are most widely used in real-time estimation [14]. The dynamic system described in Eq. (11) is not far from absolutely linear as the spacecraft is rotating slowly compared to the sampling rate, which is required by assumptions made leading to Eq. (6) and Eq. (10). In this section, we propose an adaptive Kalman filter to find the best estimate of ωs t. A. Kalman Filter Equations
1. Prediction Equation The Kalman filter equations can be divided into two parts, namely the prediction equation and the update equation. For all the available stars in the star image captured at the time of t, we construct the prediction equation as Xtk I 3×3 Xtk−1 W 1 tk−1 ;
where Xtk ωs t ω1 t ω2 t ω3 t T ; Xtk and Xtk−1 are the states of the Kalman filter associated the available stars marked k and k − 1 at the time of t, respectively; W 1 tk−1 is the process noise. For different available stars in the same star image, their states all denote the angular velocity associated with the star image. That is, the angular velocity is the same for all the measurements in the same star image frame. Therefore, the transfer matrix from k − 1 to k moment in Eq. (13) is the identity matrix, and the process noise W 1 tk−1 is ideally zero. However, using a Kalman filter without noise in the dynamic update equation is usually considered to be risky as it leads to a null value of the Kalman gain. Therefore, we assume that the covariance matrix of the process noise Q1 is in standard form with ξI 3×3 instead of a zero matrix, where ξ is a very small positive quantity. 2. Measurement Equation According to Eq. (11), the measurement equation is in the form Z1 tk H 1 tk Xtk V 1 tk ;
k1
Once the star-tracker-referenced angular velocity ωs is achieved, the body angular velocity of the spacecraft can be derived as the transform matrix from the spacecraft platform frame to the star tracker frame is determined, even on orbit. 3. Angular Velocity Estimation with Kalman Filter
Once all available star measurements have been determined for the star image captured at the time of t, the angular velocity ωs t associated with their measurements can be estimated by using Eq. (12). However, the noise affects the estimation accuracy. The Kalman filter is a recursive algorithm that 3592
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(13)
(14)
where Z1 tk H 1 tk vtk ×
vtk − vt − Δtk ; Δt
2
6 1 p 6 2 2 24 x tk y tk f
0
−f
(15)
ytk
3
7 0 −xtk 7 5: (16) −ytk xtk 0 f
V 1 tk is the measurement noise vector given by
V 1 tk ωs t × δvtk
1 δvtk − δvt − Δtk ; (17) Δt
where δvtk is the noise of the star tracker measurement vector vtk. Then the following expressions of V 1 t can be derived: 1 Efδvtg−Efδvt−Δtg Δt ωs t×Efδvtg; (18)
EfV 1 tgωs t×Efδvtg
Obviously, the expressions of V 1 t can be ignored compared with the standard deviation generally. Thus, we can assume that V 1 t is simply zero mean white noise. Note that the sensitivity matrix H 1 tk is a singular matrix. It implies that one of the rows is a linear combination of the other two rows and that no additional information is available in a single-star measurement. Hence, to remedy this ill-conditioned problem, the row corresponding to the star tracker boresight axis measurement is left out because it is less sensitive [15]. The modified measurement equation is given as
EfV 1 t − EfV 1 tgV 1 tk − EfV 1 tgT g
Z2 tk H 2 tk Xtk V 2 tk ;
EfV 1 tV 1 tT g − EfV 1 tgEfV 1 tT g where
ωs t×EfδvtδvtT gωs t×T − ωs t×EfδvtgEfδvtT gωs t×T
2 EfδvtδvtT g: Δt2
(19)
Generally, the refresh rate of the modern star tracker can be up to 10 Hz, whereas the allowable maximum angular velocity is fairly low. Taking the nearly Earth-pointing spacecraft for example, the body angular velocity rotation is about 0.0011 rad∕s. Thus, ‖ωs t‖Δt ≪ 1 for all k. Then the following inequalities are true: ‖ωs t×‖ × ‖EfδvtgEfδvtT g‖ × ‖ωs t×T ‖
(20)
and ‖ωs tΔt‖ × ‖Efδvtg‖ Δt q 1 EfδvtδvtT g; ≪ Δt
‖EfV 1 tg‖ ≤
(21)
By using Eq. (20), the last term in Eq. (19) dominates the first term on the right-hand side. Therefore, EfV 1 t − EfV 1 tgV 1 t − EfV 1 tgT g ≈
2 EfδvtδvtT g: Δt2
−xtk 1 Z2 tk p x2 tk y2 tk f 2 −ytk −xtk−1 1 − p ; x2 tk−1 y2 tk−1 f 2 −ytk−1 1 0 −f ytk p H 2 tk : x2 tk y2 tk f 2 f 0 −xtk
(25)
(26)
V 2 tk is the measurement noise vector. Similar to V 1 tk , we can assume that V 2 t is also zero mean white noise. B. Adaptive Filter Approach
≤ ‖ωs t×‖ × ‖EfδvtδvtT g‖ × ‖ωs t×T ‖ ‖ωs tΔt‖2 × ‖EfδvtδvtT g‖ Δt2 1 ≪ 2 ‖EfδvtδvtT g‖ Δt
(24)
(22)
According to Eqs. (21) and (22), we can obtain q ‖EfV 1 tg‖≪ EfV 1 t−EfV 1 tgV 1 tk −EfV 1 tgT g:
The conventional Kalman filter methodology hinges on prior knowledge about statistical characteristics of noises. However, many factors, such as instrument aging and temperature effects, affect the detector’s noise and camera parameters in orbit [16]. Although the camera parameters can be calibrated in real time while in orbit, the measurement noise in Eq. (24) is inevitably changed. Therefore, the strategy of using an adaptive Kalman filter is imperative. The Sage—Husa adaptive filter is one of the powerful tools that are most widely used in real-time estimation when the statistical characteristics of noises are unknown [17]. According to the Sage— Husa adaptive filter, the estimation approach for the angular velocity can be designed as 8 > Pt−k Ptk−1 Q1 > > > > > εk Z2 tk −H 2 tk Xtk−1 > > > > T − T > > < Rtk 1−dk Rtk−1 dk fεk εk −H 2 tk Ptk H 2 tk g ; K k Pt−k H 2 tTk H 2 tk Pt−k H 2 tTk Rtk −1 > > > Ptk I 3×3 −K k H 2 tk Pt− > > k > > > > Xt Xt K ε k k k k−1 > > > : d 1−b∕1−bk1 k
27
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where Rtk and Ptk are the covariance matrixes of measurement noise and a priori estimate error associated with the available stars marked k in the star image captured at the time of t, respectively. K k is the gain matrix of the Kalman filter. εk is the measurement innovation. b is the forgetting factor (0 < b < 1), usually ranging from 0.95 to 0.99. Using the forgetting factor can limit the memory length of the filter, which could enhance the effect of the newly observed data on the present estimation, making new data play a major role in the estimation and letting the old data be forgotten gradually. In this filter approach, we have supposed that the measurement noise V 2 t is zero mean white noise for simplicity. This hypothesis is reasonable as mentioned previously. In Eq. (27), Xt0 2Xt − Δt − Xt − 2Δt and Rt0 Rt − Δt, where Xt − Δt and Xt − 2Δt are the estimated results of angular velocity at the times of t − Δt and t − 2Δt, respectively. Rt − Δt is the estimated measurement noise at the time of t − Δt. The initial value of the covariance matrixes of a priori estimate error Pt0 can be defined as Pt0 EfXt0 − ωs tT Xt0 − ωs tg:
(28)
ϖ _
ωs t − Δt − ωs t − 2Δt : Δt
(32)
By using the angular velocity acceleration probabilistic model, we have σ 2i
ω_ 2i 1 4pMi − p0i : 3
(33)
The parameters ω_ Mi, pMi , and p0i are considered as tuning parameters. As is customarily done, they are selected before the mission by experience with real and simulated data. Estimation Scheme
The angular velocity estimation scheme with the adaptive Kalman filter is shown in Fig. 2 and consists of six steps, as follows.
Xt0 −ωt2Xt−Δt−Xt−2Δt−ωs t Xt−Δt−Xt−2Δt Δt−ωs t Δt _ s Δt−ωs toΔt2 Xt−Δt ω
Xt−Δt
Xt−Δt−ωs t−Δt _ ωs t−Δt ϖΔt−ω s t 2 _ s − ϖΔtoΔt _ ω 0 _ s − ϖΔto _ Δt2 Xt−Δt−ωs t−Δtω
(29)
where oΔt2 and o0 Δt2 are higher-order minims. Then, by substituting Eq. (29) into Eq. (28), we obtain Pt0 EfXt0 − ωs tT Xt0 − ωs tg
1. Initially, the star spot locations are achieved by usual centroiding techniques, and then the guide star information—that is, star reference vector and magnitude—are known by using the algorithm for full-sky star identification under initial attitude establishment (IAE) mode. 2. Acquire the stars appearing in both the previous and current frame by the recursive star centroiding and identification algorithms under the tracking mode (see [12] for details), and then mark them as k, k 1; 2; …; n. 3. In the first image frame under the tracking mode, the initial value of angular velocity for the adaptive Kalman filter is achieved by the leastsquares approach as shown in Eq. (34). Then, the initial values of R0 and P0 according to Eqs. (35) and (36) are obtained respectively: n −1 1 X T X0 − vt0 Δtk × vt0 Δtk × Δt k1
EfXt − Δt − ωs t − Δt Xt − Δt T
2 _ s − ϖ _ s − ϖgΔt _ T ω _ − ωs t − Δtg Efω
Pt − Δt Δt2 Λ;
(31)
We have supposed that the angular velocity acceleration probabilistic model is similar to the Singer _ i g3i1 can be (1) equal model [18]: the components fω _ Mi _ Mi with probability pMi ; (2) equal to ϖ _ −ω to ϖ _ ω with probability pMi ; (3) equal to ϖ _ with probability p0i ; or (4) uniformly distributed over the interval _ Mi ϖ ϖ _ −ω _ ω_ Mi with the remaining probability mass. Here, ϖ _ is defined as
C.
In Eq. (28), Xt0 − ωs t can be expressed as
≈Xt−Δt−ωs t−Δtω_ s − ϖΔt; _
Λ diagfσ 21 ; σ 22 ; σ 23 g:
(30)
×
n X
vt0 Δtk ×T vt0 k ;
(34)
k1
where Pt − Δt is the covariance matrix of a priori estimate error for the star image captured at time t − Δt. For simplicity, a decoupled model is chosen for the three components in Λ, namely 3594
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R0 Z2 t0 Δtk − H 2 t0 Δt1 X 0 × Z2 t0 Δtk − H 2 t0 Δt1 X 0 T ;
(35)
Fig. 2. Scheme of the angular velocity estimate with the adaptive Kalman filter. −1 P0 H 2 t0 ΔtT1 R−1 0 H 2 t0 Δt1 :
(36)
4. Estimate the angular velocity by the designed adaptive filter as follows: (4.1) For the star marked k, the angular velocity and the characteristics of effective measurement noise are updated by using Eq. (27). (4.2) If k < n, letting k k 1, the loop will start ^ s t Xtn , then again from step 4.1. If k n, let ω ^ s t as the best estimation of the angular output ω velocity for the current frame at the time of t, and the loop will go to step 5. 5. In the filter reinitialization step, let ^ s t − ω ^ s t − Δt, Rt Δt0 Rtn Xt Δt0 2ω and Pt Δt0 Ptn Δt2 Λ, where Pt Δt0 and Rt Δt0 are the initial values of adaptive Kalman filter for the next frame; Ptn , and Rtn are the outputs of step 4; and Λ is obtained according to Eq. (30). 6. Once the filter has been reinitialized, let t t Δt, input the new star image frame captured at the time of t, then acquire the available stars similarly to those obtained in step 2, and the loop will start again from step 4.
In the simulation, the designed focal length, field of view, and star magnitude limit of the star tracker are 105.75 mm, 5.1°3.6° × 3.6°, and 8.0 mag (magnitude) respectively; and the resolution of the CCD camera is 512 × 512 pixels with pixel pitch of 13 μm × 13 μm. The star tracker data rate is 10 Hz. We supposed that the random noise of the star centroid and the calibrated effective focal length (EFL) are 0.1 pixels in 1σ and 105.72 mm (Δf 0.03 mm), respectively. The satellite orbit elements in the simulation are (6731566.67 m, 1.52 × 10−3 , 87.20°, 291.42°, 71.34°, 290.23°), which are the orbit parameters of the challenging minisatellite payload (CHAMP) satellite. The exact position and speed of the satellite platform are (2463480.91 m, −6258628.28 m, 165616.01 m) and (277.21 m∕s, 324.33 m∕s, 7687.23 m∕s), respectively, at the initial time t 0. The true values of the three-axis body angular velocity are shown in Fig. 3 over 6500 min. A more complete discussion for the simulation to continuously acquire the synthetic data of the star projection can be found in [19]. The acceleration probabilistic model parameters _ Mi 10−6 rad∕s2 , pMi p0i in Eq. (33) are set as ω 0.001 for all three axes. The estimated results of the least-squares calculation according to Crassidis’s approach using a backward difference described as Eq. (12) and the adaptive Kalman filter are plotted in Figs. 4 and 5, respectively. According to the results, the estimations of angular velocity achieved by the Kalman filter seem to be much better than that of the least-squares approach as the noise can be filtered out by the Kalman filter. In the least-squares calculation according to Crassidis’s approach, the random noise of estimated three-axis angular velocity is 0.16; 0.16; 6.24 × 1.0−4 rad∕s in 1σ, while the designed Kalman filter can improve the final threeaxis angular velocity accuracy to 0.74; 0.78; 7.92 × 1.0−6 rad∕s in 1σ.
4. Simulation and Experimental Testing A.
Simulation and Results
The real-time simulation is adopted to test performance of the proposed angular velocity estimation approach, in which the synthetic data of the star projection in the star tracker frame can be acquired continuously when the satellite is on orbit.
Fig. 3. True values of the three-axis body angular velocity. 1 June 2012 / Vol. 51, No. 16 / APPLIED OPTICS
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Fig. 6. One of the night-sky images with highlighted centroid locations of the available stars.
Fig. 4. Estimation errors with the least-squares approach.
B.
Experimental Testing with Night-Sky Observation
To further show perfect performance of the proposed approach, images are taken with a high-quality astronomical camera in a single night. After the images have been processed and centroided and stars have been identified, the estimation methods are run on each image to measure the angular velocity of the Earth in the star tracker frame. 1. Experimental Setup The parameters of the camera are described in section 4.A, and the download of a full-frame image requires approximately 0.33 s. After calibration by the optical laboratory method proposed in [20], the deviation of the measured interstar angle is less than 2.5 × 10−5 rad including the star catalog errors.
The camera is fixed on a tripod equipped with a vibration absorption setup. In order to reduce the influence of the atmosphere, the optical axis is pointed to the zenith as far as possible during the star observation, in which the maximal difference angle of the star direction in the field of view (FOV) with the zenith direction is about FOV∕2. 2. Experimental Results A total of 1315 images were taken in a single night using the equipment described above. The rotational angular velocity of the Earth is about 7.292e−5 rad∕s, while it is about 0.0011 rad∕s for a nearly Earthpointing spacecraft. Without loss of generality, the interval time Δt between two successive frames is chosen as 1.5 s during the experiment. Thus, the rotational angle of the Earth between two successive image frames is approximately equal to that of the nearly Earth-pointing spacecraft when the refresh rate is 10 Hz. One of the night-sky images used for testing the estimation approach is shown in Fig. 6, in which the corresponding locations of the available stars Table 1.
Star Identification Results in Epoch 2000 Reference Frame
Right Centroid Ascension Declination Number Location (pixels) (deg) (deg) Magnitude
Fig. 5. Estimation errors with the adaptive Kalman filter. 3596
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1 2 3 4 5 6 7 8 9 10 11 12 13 14
460.93 135.10 472.09 177.76 236.98 460.95 227.50 278.56 70.87 258.47 187.90 504.46 162.10 334.77
420.89 485.19 129.76 314.22 324.39 65.38 60.18 482.72 242.77 437.25 461.10 445.94 54.65 337.24
13.99379 13.40406 11.93758 12.29037 12.56136 11.44380 10.62070 13.85509 11.40484 13.45690 13.39934 14.31337 10.35581 12.97657
27.20938 29.48931 26.29165 28.71927 28.36898 26.17304 27.64291 28.55944 29.18968 28.55821 29.08023 26.99953 28.04264 27.77948
6.08 6.72 6.89 7.183 7.22 7.44 7.2 7.2 7.73 7.34 7.32 7.76 7.86 7.83
increases the accuracy remarkably, and convergence is achieved after about 90 s (about 60 images). The rotational angular velocity of the Earth can be regarded as a constant. Therefore, the three-axis angular velocity in the star tracker frame is invariable as the installation azimuth angle of the star tracker is not changed during the experiment. Thus, the acceleration probabilistic model parameters in Eq. (33) are all set to zero. To show the performance of the estimation approaches further, we investigate the statistical characteristic of the estimated results. Table 2 shows the mean and standard deviation of the estimated results of the last 300 images. According to Table 2, the random noises of estimated three-axis angular velocity have been reduced from 0.049; 0.049; 1.854 × 1.0−4 rad∕s to 0.183; 0.158; 3.324 × 1.0−7 rad∕s in 1σ if the designed Kalman filter is adopted. Fig. 7. Angular velocity estimation results with the least-squares approach.
Fig. 8. Angular velocity estimation results with the adaptive Kalman filter.
are identified successfully. The star identification results obtained with the grid algorithm (see [21] for more details) are shown in Table 1. Figures 7 and 8 show the results for the angular velocity estimates of the Earth from the experimental data. According to the results as shown in Fig. 7, there are large variation and uncertainty for the least-squares estimation given by Eq. (12), especially in the roll axis. Obviously, the designed Kalman filter Table 2.
Least-squares Adaptive Kalman filter
5. Conclusions
The approaches with least squares and the adaptive Kalman filter are presented for estimating angular rate directly from star tracker measurements. In these approaches, only knowledge of the star tracker vector measurements and sampling interval are required to derive an angular velocity estimate, while the attitude and star reference vectors are not required to be known. To confirm the proposed estimation approaches, the simulations based on the orbit data of the CHAMP satellite and an experimental test with night-sky observation are performed. Both results have demonstrated the proposed approaches perform well in terms of accuracy, robustness, and performance. In contrast with Crassidis’s approach, the main advantage of this approach is that the adaptive Kalman filter is adopted to filter out noises without the information of the dynamic model and inertia dyadic. The Kalman filter is a recursive algorithm that has the advantage that a large amount of information does not need to be stored. The angular rates are updated and the covariance is propagated by each guide star. Once the update is performed, the data of the star measurements at the previous frame can be discarded. Therefore, compared with the least-squares method in Crassidis’s approach, no computational cost has been increased to implement the Kalman filter. Another advantage is that we redefine the solution of Crassidis using a backward difference instead of a forward difference. Thus, the estimation delay in Crassidis’s approach can be avoided. However, from a numerical point of view, estimating a derivative at the time of t should be done by exploiting data from both the past and future of t (as in smoothing or fixed-lag filtering). Because this is a causal
Statistical Analysis of the Last 300 Images
Mean (rad∕s)
Standard Deviation (rad∕s)
5.768; −2.673; 3.337 × 1.0−5 5.781; −2.695; 3.406 × 1.0−5
0.049; 0.049; 1.854 × 1.0−4 0.183; 0.158; 3.324 × 1.0−7
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approach, the estimation of the derivative using onesided data (just from the past) will typically lead to worse results than in noncausal approaches. Therefore, exploiting data in a fixed-lag filtering approach, we probably obtain better results at the cost of an estimation delay. The direction vector measurement is assumed to shift with a certain regular velocity in the interval between successive frames similarly to Crassidis’s approach. The initial value of angular velocity for the Kalman filter is obtained based upon the difference approximation according to the estimated results of the former two frames. These may be slightly different from the actual situation and results in higher system error if the refresh rate of the star tracker is relatively low. Fortunately, the refresh rate of a modern active-pixel-sensor star tracker has been higher than 10 Hz, whereas the allowable maximum angular velocity is fairly low. In view of these considerations, this difference approximation is advisable, and a guide star will be captured in many successive frames (typically several hundred frames) before it leaves the FOV [22]. Besides, the available stars that enter or leave the FOV must be taken into account. If a star leaves the FOV at the time of t, then this star should not be used. This can be easily employed using simple tracking logic, such as computing inner star angles between successive body measurements in [23]. This research was jointly supported by the Graduate Innovation Fund of the National University of Defense Technology (2010B100202), by Hunan Provincial Innovation Foundation for Postgraduate (CX2010B005), partially supported by the National Natural Science Foundation of China (NSFC) and the School Advance Research of the National University of Defense Technology (JC11-02-22). References 1. M. A. Paluszek, J. B. Mueller, and M. G. Littman, “Optical navigation system,” in AIAA Infotech at Aerospace 2010, April 20, 2010–April 22, 2010 (American Institute of Aeronautics and Astronautics, 2010). 2. H. Leeghim, Y. Choi, and B. A. Jaroux, “Uncorrelated unscented filtering for spacecraft attitude determination,” Acta Astronaut 67, 135–144 (2010). 3. B. N. Agrawal and W. J. Palermo, “Angular rate estimation for gyroless satellite attitude control,” in AIAA Guidance, Navigation, and Control Conference and Exhibit 5–8 August 2002 (AIAA, 2002). 4. M. D. Shuster, “A survey of attitude representations,” J. Astronautical Sciences 41, 439–517 (1993).
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