Design of on-board calibration methods for a digital sun sensor based on Levenberg–Marquardt algorithm and Kalman filters

Ali RAHDAN, Hossein BOLANDI, Mostf ABEDI

a Department of Electrical Engineering, Iran University of Science & Technology, Tehran 1684613114, Iran

b Department of Electrical Engineering, Shahid Beheshti University, Tehran 1983969411, Iran

KEYWORDS

Abstract Digital sun sensor is one of the most important sensors used in the Attitude Determination System (ADS) of the satellite. Due to the harsh environmental conditions that exist in the space,various distortions may occur in the sun sensor optical system that lead to the reduced accuracy of this equipment. So, it is necessary to recalibrate the optical parameters of the aforementioned sensors. For this purpose, first a novel attitude independent error model is proposed for the SS-411 sun sensor that includes the central point of the CCD array, installation error, filter thickness and sensor misalignment. So, the mutual interfaces between the sensor parameters are considered in the developed model.In order to extract the sensor parameters,a nonlinear optimization technique called the Levenberg–Marquardt is applied to the developed model as a batch algorithm. In addition, the Extended Kalman Filter (EKF) and the Unscented Kalman Filter (UKF)have been utilized as sequential strategies.It will be shown that by considering a worst case of variation amount for sensor parameters, an accuracy improvement of about 17° is achieved by the developed calibration algorithms. Comparison between the developed algorithms represents that UKF has higher accuracy, shorter time convergence but higher computational load.

1. Introduction

The sun sensor is one of the most important sensors used in the attitude determination and control system of the satellite. It senses the presence of the sun and determines its direction.1Due to the diversity of the design of the sun sensors,including analog2,3or digital types,4,5each of them has its own parameters. Due to instrument aging, intense vibration during the launching process as well as variable environments during operation, the mentioned parameters may change, leading to mismatches between the default and in-orbit values.2So, to obtain precise sun vector calculation, sun sensors should be recalibrated in orbit.

Two types of approaches such as attitude dependent and attitude independent have been developed for on-board calibration of the satellite sensors. In attitude dependent approach, attitude matrix is used for extracting the sensor parameters.6However, the attitude estimation error affects the calibration accuracy.1Therefore, the focus of researchers has been on the attitude independent strategies. These approaches are divided into offline and online categories. In offline methods, an entire set of data must be stored to determine the unknown parameters.In online methods,parameters are estimated with each sample of the sensor output.7,8

So far, various offline methods have been applied in some attitude determination sensors. The magnetometer bias is obtained in Ref. 9 by minimizing the weighted sum of the squares of the residuals. Alonso and Shuster10,11proposed a new algorithm called two-step that is an extension of the Gambhir’s algorithm.12It was also investigated by Alonso and Shuster13that the two-step algorithm can be developed to estimate scale factors and non-orthogonality corrections.Samaan et al.14introduced two calibration methods to obtain the focal length and principal point offsets and the focal plane distortions for a star camera.The developed algorithms ignore the mutual relation between parameters that causes the error in calibration. A recursive least squares algorithm was introduced by Woodbury and Junkins15to estimate the optical parameters.This method only estimates the optical parameters of star sensors and ignores the lens distortion estimation,which reduces the position accuracy of the star centroids.Zhou et al.8investigated a nonlinear optimization technique to simultaneously estimate the optical parameters and star sensor distortion. This algorithm takes into account the mutual effect between the optical parameters and the lens distortion,leading to more accurate parameter estimation for star sensors.Wang et al.16proposed an improved version of the least square with which the optimal principal point, focal length, and the high-order focal plane distortion can be obtained. An inorbit calibration model has been introduced by Fa and Peng17to improve the sun sensor accuracy and the least square regression algorithm has been applied to this model to optimize the internal and external parameters. Some online methods have also been introduced. The Extended Kalman Filter (EKF)and sequential center solution algorithm have been used by Crassidis et al.18to establish the magnetometer bias,scale factor and non-orthogonality. Juang et al.19combined the satellite position equations and the magnetometer model to estimate the position vector and magnetometer bias and scale factor.Z.Zhang et al.20have proposed a new differential value estimation method to obtain the magnetometer bias. In order to improve the in-orbit measurement accuracy of star sensors,the effects of image plane rotary error, image plane tilt error and distortions of optical systems were studied.21H. Zhang et al.22considered the simplified back propagation neural network for focal length and principal point estimation,and then the UKF was adopted for the precise calibration of all mentioned parameters.

Earlier studies show that there is not a comprehensive and all-purpose method for digital sun sensors,while the optical or mechanical parameters of this sensor are quite likely to change due to the space environment.It can affect the sun sensor field of view or accuracy, leading to decrease of satellite attitude determination accuracy in normal or imaging modes. Accordingly, an attitude independent methodology is investigated in this paper for these sensor types for which purpose the SS-411 digital sun sensor has been selected with wide heritage in different space missions. Based on the idea developed by Samaan et al.,14Woodbury and Junkins,15Wang et al.16and Z. Zhang et al.22for the star cameras, we have suggested a novel error model for the selected sun sensor which includes the central point of CCD array, the angle between the linear array and the sensor frame and the filter thickness. Furthermore, unlike the mentioned methods that only consider the optical model of the sensor,the misalignment,the internal sensor distortions and the output white noise are also involved in the introduced model.Therefore, since the minimal simplification has been made in the aforementioned model, the mutual influences between different optical and mechanical parameters have been taken into account, which can yield more accurate calibration process.

In this paper, three autonomous in-orbit calibration methods (including online and offline strategies) are adopted in the aforementioned error model to estimate the sun sensor parameters. For offline case, based on the research done by Zhou et al.,8a nonlinear optimization technique called the Levenberg–Marquardt is developed, which simultaneously extracts the real values of the central points of the CCD array, filter thickness and the CCD array distortions using a batch of data.According to Zhou et al.8and Samaan et al.,14this algorithm is more accurate than the Newton-Gauss algorithm and is more resistant to the initial conditions. Also, the accuracy of this algorithm is higher than the recursive least squares.8,15Therefore, this algorithm is used in high-precision offline application. About the online case, the EKF and UKF have been applied for which only the developed model of the sun sensor is utilized.So,it is not required to combine their formulation with the satellite dynamic or kinematic equations that we can see in several references like Ref. 19. The set of the mentioned strategies provide an independent package that covers all satellite operating modes, whether data storage or online. They can simultaneously treat most of the mechanical and optical parameters of the digital sun sensor,leading to sensor field of view correction and more accurate estimation of the sun vector. Since the developed error model requires the information of light spots of CCD array at any moment,a sensor output generation scenario has been designed according to which the active pixels are obtained using the sun vector location in the inertial frame.

The main contributions of this study are: (A) based on the author investigations, a similar approach has not been reported so far, which can accurately capture the internal parameters of a complex sun sensor like SS-411. In fact, more research is focused on magnetic or star sensor, while occurrence of some distortions in the optical system or mechanical parts of sun sensors can also have profound effects in the satellite operating modes.(B)a novel error model is introduced for the sun sensor involves most of optical and mechanical parameters including the central points of the CCD array, the angle between the linear array and the sensor frame, the filter thickness, the sensor misalignment, the internal sensor distortions and the output white noise. (C) Three types of on-board calibration algorithms (including batch and sequential) are adopted for the developed error model which can accurately obtain all internal distortions of the digital sun sensors after the satellite launch. Note that the introduced philosophy can be developed for other sun sensor types. (D) A sensor output generation scenario is suggested according to which the sun vector location in the inertial frame is related at any moment to the active pixels of the CCD array.

The paper is organized as follows. In Section 2, the error sources of SS-411 digital sun sensor and its mathematical model are expressed. Section 3 presents the output generation scenario of the sensor. Section 4 proposes the sun sensor calibration methods. In Section 5, the simulation results are detailed. Ultimately the paper is concluded in Section 6.

2. Digital sun sensor model

2.1. Optical parameters of digital sun sensor

As mentioned before, the SS-411 digital sun sensor has been selected because of high accuracy and its wide heritage.A schematic of the sensor optics has been shown in Fig.1.The front mask on the sensor allows light to enter,and then it is refracted through the filter, separating the mask from the linear array.Each slit produces bright line on the array plane. The placement of these bright lines depends on the mask feature, captured inDxandDy, the filter thicknesshand the angle of sun radiation. The lines of illumination create regions of high response where they intersect the pixel array. According to Fig.2, the position and orientation of the detector will determine which pixels are active.Array placement can be specified by three simple parameters. These parameters include the coordinates of the first pixel (central point) （ρx，ρy）, and the angle between the linear array and thexaxis ψ. The center of these light spots is determined by using the peak position algorithm. Finally, the sun vector is given by23

where

Fig.1 Geometric model of sensor.

Fig.2 Detailed view of focal plane geometry.

wdenotes the sun vector, ΔXis the size of each pixel,nglassis the filter fraction factor and （mx1，my1，mx2，my2） represent centers of the light spots.

2.2. Developed model for digital sun sensor

Considering Eq. (1), the attitude dependent mathematical model of the SS-411 digital sun sensor is as follows:

If the attitude matrix of the orbital frame with respect to the body-fixed frame is constant,we can use the fact that inner product for ideal unit sun vectors in orbital frame and sensor frame are the same, which is based on Eq. (4), and it can be expressed mathematically as

3. Output generation scenario of SS-411 digital sun sensor

In the previous section, an attitude independent model based on Eqs. (2), (3) and (5) is obtained. As we can see in these equations, it is required to obtain the average of centers of the light spotsmx=（mx1+mx2）/2 andmy=（my1+my2）/2 at any moment to calculate the sun vector measured by the sensor. Therefore, an output generation scenario is developed in this section according to which the sun vector in the inertial frame is related to the above light spots in the sensor array.Fig.3 shows different steps to derive the aforementioned relation.In order to generate light spots of the SS-411 sun sensor,we need to know the attitude matrix between the reference and the sun sensor frame,and the reference vector from the satellite toward sun.The mentioned attitude matrix is given by the following equation:

In order to generate a reference vector from the satellite toward the sun (ssat-sun), first, the reference vector from the earth toward the sun (searth-sun) and the reference vector from the earth toward the satellite(searth-sat)are calculated and then by subtracting them from each other, the mentioned vector is obtained.The first vector(searth-sun)is derived through the relations governing the science of astronomy.1According to Fig.3,the second vector(searth-sat)is also calculated through modeling the orbital dynamics of the satellite.

wherew1,w2andw3are the elements ofwandn′is the light spots noise. Finally, valid positions are filtered due to the size of the linear array.

4. Design of proposed on-board calibration methods

4.1. Developed model for digital sun sensor

In the offline calibration process, the sun reference vectors[s1s2...sN]Tand the coordinates of the light spots[（mx，my）1（mx，my）2... （mx，my）N]Tare received as inputs for a given time period after storage. Then, by using the nonlinear least squares algorithm, the optical parameters of the sensor are derived.The maximum likelihood estimation can be obtained through minimizing the following cost function:

Fig.3 Output generation scenario of SS-411 digital sun sensor.

whereF1j（x） is obtained by replacingi=1 in the right side of Eq. (5). So, we have a nonlinear minimization problem which is solved using the Levenberg–Marquardt algorithm.The solution process requires providing initial guess values of ρx, ρy,h, and ψ. Usually, parameters obtained from on-ground calibration are selected as initial guess of the sensor parameters. Fig.4 shows the calculation flowchart of this algorithm. According to this figure, the following steps should be performed:

(1) Selecting parameters obtained from the on-ground calibration as initial guess(x0),and receiving the coordinates of the light spots [（mx，my）1（mx，my）2...（mx，my）N]Tproduced by the output generation scenario and the sun reference vector [s1s2...sN]T.

(2) Calculation of Jacobian matrix due toNmeasurement data as follows:

Fig.4 Calculation flowchart of Levenberg–Marquardt algorithm.

(3) Calculation of the measurement vector and the noise covariance matrix as

(4) Update parameters as follows:

After performing and iterating the steps(2)-(4),the optimal solution for calibration parameters is obtained. Note that the detailed calculation process is not described here for brevity.

4.2. Kalman filter based online methods

In the online calibration process, the sun reference vectorskand coordinates of the light spots （mx，my）kare sampled at each sample time. Then, the optical parameters of the sensor are obtained using the extended Kalman filter or unscented Kalman filter.

4.2.1. Extended Kalman filter based online method

In order to apply the extended Kalman filter, the state equation and the measurement equation must be specified. In this regard, sensor parameters are considered as states. Due to the constant values of sensor parameters, the state equation and the measurement equation are as follows:

Note that the measurement Eq.(19)has been developed by inserting Eqs.(1)and(11)into the attitude independent model represented in Eq. (5). Fig.5 shows different steps of the sun sensor in-orbit calibration based on EKF. According to this figure, the following steps should be performed:

Fig.5 Extended Kalman filter algorithm.

(1) The initial conditions for the states (^x0) and the state covariance matrix (P0) are chosen.

(2) Derivation of the Jacobian matrix as

(3) Calculation of measurement value and measurement noise variance as

(4) Calculation of Kalman filter gain as

(5) Update of state vector and covariance matrix as follows:

Steps (2) to (5) are repeated until the algorithm converges to the desired value.

4.2.2. Unscented Kalman filter based online method

The state and the measurement equations for applying this algorithm are in accordance with Eqs.(18)and(19).Fig.6 shows different steps of the sun sensor in-orbit calibration based on UKF.According to this figure,the following steps should be performed:

(1) Choosing the initial conditions for states (^x0) and the state covariance matrix (P0)

(2) Determination of the sigma points as follows:

(3) Obtaining the weight coefficients:

where λ denotes the scaling parameter which satisfies

(4) Propagation of sigma vector through the equation below:

(5) Approximation of the mean and covariance using a weighted sample mean and covariance of the posterior sigma points as follows:

(6) Propagation of the nonlinear measurement equation as

(7) In the measurement update stage of the UKF, the measurement noise covariance and the cross correlation matrix are calculated as follows:

(8) Calculation of the Kalman filter gain as

Fig.6 Unscented Kalman filter algorithm.

(9) Update of the posterior states and posterior covariance matrix as follows:

Steps (2) to (9) are repeated until the algorithm converges to the desired value.

5. Simulation

In this section, performance of the proposed algorithms is evaluated in terms of accuracy, convergence time, amount of storage and calculation time for a digital sun sensor mounted on a low earth orbit satellite body.The satellite orbital parameters are considered according to Table 1.Also,the installation error between the body and sensor frames is as follows:

Table 1 Satellite orbital characteristics.

Table 2 SS-411 digital sun sensor characteristics.

Fig.7 Footprint of sun spots on linear array.

The SS-411 digital sun sensor characteristics have been also given in Table 2.Fig.7 shows the footprint of the sunspots on the linear array over a period of the satellite orbit.These spots are used as inputs in the calibration algorithms.

5.1. Levenberg–Marquardt algorithm

In this section, the performance of the Levenberg–Marquardt algorithm is evaluated.This algorithm requires the initial conditions and a number of data in a given time period.Hence,the initial conditions are considered as (-5.7 mm, –5.7 mm) for the central point, 4 mm for the filter thickness and 45°for the angle between thexaxis and the linear array. Also data is collected for 4 days with sampling time of 20 s. By applying 0.2 pixels of white noise and executing this algorithm 100 times, Figs. 8 and 9 are obtained. As it can be observed, the algorithm converges after four steps.The obtained values after convergence have been given in Table 3. This table shows parameters variations about 12.30% between the noncalibrated (on ground) values and on orbit ones. According to this table, the absolute error of the central point is (0.022,0.003) mm, the absolute error of the filter thickness is 0.016 mm, and the absolute error of the angle between thexaxis and the linear array is equal to 0.15°. Fig.10 shows the angle difference (θ) between the true sun vector and the calibrated sun vector at any light spot position.It is deduced from this figure that maximum difference occurs in the neighborhood of (mx=100 pixels,my=150 pixels) with the value of 0.40°.

Fig.8 Central point length obtained by Levenberg–Marquardt algorithm.

Fig.9 Filter thickness and array angle obtained by Levenberg–Marquardt algorithm.

Table 3 Calibration results of Levenberg–Marquardt algorithm.

Fig.10 Angle difference (θ) between true sun vector and calibrated sun vector resulted in by Levenberg–Marquardt algorithm.

Table 4 True sun vector, calibrated sun vector, and noncalibrated sun vector in sensor coordinates(Levenberg-Marquardt).

Therefore, considering spots coordinates as(mx=100 pixels,my=150 pixels) and a variation amount of about 12.30% for sensor parameters (in worst case), the angle between the true sun vector and the non-calibrated sun vector is equal to 17.6°. Also the angle between the true sun vector and the calibrated sun vector is 0.4°.Therefore,the calibration process increases the sensor accuracy as 17.2°.Aforementioned vectors are shown in Table 4.Note that due to consideration of the large variation of the sensor parameters in simulation, the angle difference between two vectors is reasonable. However,in actual situation, variation of sensor parameters may be small, which leads to small angle difference between two vectors. Therefore, the obtained results indicate that the developed algorithm also has acceptable performance in strict conditions.

Tables 5 and 6 show the effect of the data collection time and the sampling time on the calibration accuracy. Based on the above tables, the accuracy of the sensor calibration is improved by increasing the number of available data (collection time). So the accuracy can be upgraded to the extent that the memory and the processor limitations allow. However,with an increase in sampling time of up to 20 s,there is no significant effect on the accuracy of the results,which is due to the slow dynamics of the sun with respect to the satellite. By increasing the sampling time from the above value,the estimation accuracy will be degraded. So, considering different aspects such as the memory limitations, the conventional telemetry rate in the space missions and other operational requirements, the sampling time of 20 s can be a suitable choice.Therefore,this sampling time is used in the calibration process.

5.2. Extended Kalman filter algorithm

In this section, the performance of the extended Kalman filter algorithm is evaluated. The initial conditions of the central point, the filter thickness and the angle between thexaxis and the linear array are selected similar to the previous section.Also the values diag（10-5， 10-5， 10-6， 10-3） and 10-4are considered respectively for the state error covariance matrix and the noise variance. The sampling time is 5 s. By applying 0.2 pixels of white noise and executing this algorithm 100 times,Figs.11 and 12 are obtained.As it is observed,the algorithm converges after 8 days (691,200 s). The obtained valuesafter convergence have been given in Table 7. According to this table, the absolute error of the central point is (0.016,0.002) mm, the absolute error of the filter thickness is 0.003 mm and the absolute error of the angle between thexaxis and the linear array is equal to 0.12°. Fig.13 shows the variation of Mean Squares Error(MSE)of the optical parameters. According to this figure, the mean squares error reaches less than 0.001 mm2after 5 days (432000 s) and its value is equal to 0.00009 mm2after 8 days (691200 s).

Table 5 Effect of data collection time and sampling time on calibration accuracy.

Table 6 Effect of data collection time and sampling time on maximum angle difference(θ)between true sun vector and calibrated sun vector.

Fig.11 Central point length obtained by EKF algorithm.

Fig.12 Filter thickness and array angle obtained by EKF algorithm.

Table 7 Calibration results of extended Kalman filter.

Fig.14 shows the angle difference (θ) between the true sun vector and the calibrated sun vector at any light spot position.It is deduced from this figure that maximum difference occurs in the neighborhood of (mx=100 pixels,my=150 pixels)with the value that is equal to 0.22°.

So, for a variation amount of about 12.30% for sensor parameters (in the worst case), the angle between the true sun vector and the non-calibrated sun vector is equal to 17.6°. Also the angle between the true sun vector and the calibrated sun vector is 0.22°. Therefore, the calibration process increases the sensor accuracy as 17.38°. Aforementioned vectors are in accordance with Table 8.

Fig.13 Variation of mean squares error of optical parameters(EKF).

5.3. Unscented Kalman filter algorithm

In this section,the performance of the unscented Kalman filter algorithm is evaluated.The initial conditions are similar to the EKF algorithm.By applying 0.2 pixels of white noise and executing this algorithm 100 times, Figs. 15 and 16 are obtained,which show a convergence time of about 4 days (345,600 s).

Fig.14 Angle difference (θ) between true sun vector and calibrated sun vector (maximum difference is 0.22°).

Table 8 True sun vector, calibrated sun vector, and noncalibrated sun vector in sensor coordinates (EKF).

Table 9 Calibration results of unscented Kalman filter.

The obtained values after convergence have been given in Table 9.According to this table,absolute error of central point is (0.005, 0.002) mm, absolute error of the filter thickness is 0.002 mm and the absolute error of the angle between thexaxis and the linear array is equal to 0.03°. Fig.17 shows the variation of the mean squares error of the optical parameters.According to this figure, the mean squares error of the optical parameters reaches less than 0.0002 mm2after 2 days(172800)and the value is equal to 0.00001 mm2after 4 days(345,600 s).Fig.18 shows the angle difference(θ)between the true sun vector and the calibrated sun vector at any light spots position.It is deduced from this figure that maximum difference occurs in the neighborhood of (100, 150) pixels with the value that is equal to 0.10°.

Fig.15 Central point length obtained by UKF algorithm.

Fig.16 Filter thickness and array angle obtained by UKF algorithm.

Fig.17 Variation of mean square error of optical parameters(UKF).

Fig.18 Angle difference (θ) between true sun vector and calibrated sun vector (maximum difference is 0.10°).

Table 10 True sun vector, calibrated sun vector, and noncalibrated sun vector in sensor coordinates (UKF).

So, for a variation amount of about 12.30% for sensor parameters (in the worst case), the angle between the true sun vector and the non-calibrated sun vector is equal to 17.6°. Also the angle between the true sun vector and the calibrated sun vector is 0.10°. Therefore, the calibration process increases the sensor accuracy as 17.5°.Aforementioned vectors are in accordance with Table 10.

The obtained results from the simulation of the proposed algorithms are summarized in Table 11. In this table, the‘‘data collection time” is the duration required for collection of the required data for offline methods and the ‘‘convergence time” is the time required for convergence of online algorithms. The ‘‘calculation time” is derived for an Intel(R) Core (TM) i7-3537U CPU @ 2.00 GHz. It is the execution time of each stage of the algorithm. Also the storage memory is the required memory for storing the input data of the algorithm. It can be inferred from this table that the UKF has better conditions in estimation accuracy of central point, filter thickness and array angle, and also it has lower convergence time than EKF. However, it suffers from the complexity in implementation and requires better processor. Note that although the Levenberg–Marquardt algorithm has high storage memory, it has its own application especially when we cannot execute the algorithms in online operation modes or there are some process limitations in real-time executions.

6. Conclusions

(1) An error model of SS-411 sun sensor was introduced,which includes the central points of CCD array, the angle between the linear array and the sensor frame,the filter thickness,the sensor misalignment,the internal sensor distortions and the output white noise.

(2) The Levenberg–Marquardt algorithm was applied to the developed model as a nonlinear optimization technique.Also,the EKF and UKF were designed as online strategies for obtaining the sensor optical parameters. It was established in the simulation results that different distortions in the internal structure of sun sensor can be accurately treated by the developed algorithms,which is due to the fact that mutual influence between different sensor elements has been taken into account.

(3) It was shown that calibrated sun vector is about more accurate than the values measured by the sensor itself.It is inferred from the comparison of different algorithms that the UKF leads to a higher accuracy in central point and filter thickness, and also with a convergence time of 4 days and no need for storage memory, it is superior to other algorithms; however, it requires more computations (better processor is required).

(4) As a future work, it is suggested that an algorithm is developed to obtain more accurate initial conditions,which leads to less convergence time.

Table 11 Results obtained by simulation of proposed algorithms.

Acknowledgement

We acknowledge Dr. Pourgholi for his valuable suggestions and guide in this research.