## Abstract

The star sensor is a prerequisite navigation device for a spacecraft. The on-orbit calibration is an essential guarantee for its operation performance. However, traditional calibration methods rely on ground information and are invalid without priori information. The uncertain on-orbit parameters will eventually influence the performance of guidance navigation and control system. In this paper, a novel calibration method without priori information for on-orbit star sensors is proposed. Firstly, the simplified back propagation neural network is designed for focal length and main point estimation along with system property evaluation, called coarse calibration. Then the unscented Kalman filter is adopted for the precise calibration of all parameters, including focal length, main point and distortion. The proposed method benefits from self-initialization and no attitude or preinstalled sensor parameter is required. Precise star sensor parameter estimation can be achieved without priori information, which is a significant improvement for on-orbit devices. Simulations and experiments results demonstrate that the calibration is easy for operation with high accuracy and robustness. The proposed method can satisfy the stringent requirement for most star sensors.

© 2017 Optical Society of America

## 1. Introduction

The development of earth observation, remote sensing and deep space exploration lead to increasing requirements for precise attitude information of a spacecraft. The star sensor is a representative attitude determination device with high accuracy, light weight and autonomy [1–5]. As a key component for the vehicle, the measurement information of a star sensor is always regarded as the attitude reference and is extremely crucial for the Guidance Navigation & Control (GNC) system. In the New Horizons spacecraft, two Galileo Avionica Autonomous Star Sensors (ASTR) are used to determine the absolute orientation [6–8]. For the SHarp Edge Flight EXperiment (SHEFEX) mission, a star sensor designed by German Aerospace Center (DLR) is used as the primary attitude sensor [9]. The Project for On Board Autonomy (PROBA) satellite, developed by ESA, carries a fully autonomous star sensor system for the inertial attitude reference [10]. The Space Technology Experiment and Climate Exploration (STECE) satellite mission also carries an ASTRO 10 star sensor from Jena-Optronik GmbH as the attitude and orbit control system (AOCS) sensor [11].

Since the star sensor is much more needed and plays a decisive role in the GNC system, its on-orbit parameters should be realized and guaranteed. In general, laboratory calibration and testification are conducted to obtain true values of all parameters [12–14]. However, with different calibration environments between ground and on-orbit, it is difficult to obtain on-orbit parameters beforehand in laboratory. Moreover, many factors, such as vibration during launching, component aging and variable environments during operation, alter the systematic errors and bring about mismatches between default parameters and on-orbit ones, leading to degraded attitude accuracy [15–17]. To guarantee and enhance the operation performance of star sensors during service, on-orbit calibration is essential and has a significant effect on the star sensor and GNC system.

The on-orbit calibration for spacecraft attitude sensor is firstly proposed by Shuster.M.D. and Oh.S.D., in which the relative alignment error is estimated. Even with roughness and imperfection, it is proved to be effective in several missions performed by NASA Goddard Space Flight Center [18]. Several other algorithms have also been studied [19–21]. However, only one key parameter was estimated in these methods and the inner parameters of star sensor are ignored.

Considering on-orbit calibration of star sensors, least square methods and Kalman filters are the most widely used approaches [22–25]. Samaan presented a focal plan distortion calibration method, in which the principal point was supposed to be constant [26]. Wang proposed an attitude-independent calibration approach with least-square estimation [27]. Zhou used a non-linear optimization technique to simultaneously obtain the star sensor parameters [28]. Liu also developed an autonomous calibration approach with modified least squares iteration algorithm combing Kalman filter [29]. Wang presented a two-step method based on a novel error model using the extended Kalman filter and least-square method [30]. With various measurement models, all methods mentioned above can achieve parameters calibration. However, the main disadvantage of these methods is that priori information, such as the starting guesses of parameters values, are needed, which means the calibration still highly depends on laboratory data. In some cases, the filter may become divergent with unsuitable initialization. More seriously, traditional calibration methods will fail in operation with no available priori information. This becomes an important constraint for on-orbit star sensors and will eventually influence the GNC system. Consequently, an autonomous on-orbit calibration method without prior information is of great significance for on-orbit star sensors accuracy.

In this paper, a novel on-orbit calibration method for star sensors combing the neural networks and unscented Kalman filter is proposed. Only interstar angles of navigation stars are needed. All parameters of the star sensor can be estimated without any priori information. This method should have the merits of self-initialized, attitude independent, easy operation, high accuracy and robustness. Simulations and experiments are conducted to show the effectiveness and superiority. The proposed calibration method has the potential to become a universal on-orbit calibration method.

## 2. Measurement model of a star sensor

#### 2.1 Basic model

The ideal measurement model for a star sensor can be described using the pinhole model [1], as shown in Fig. 1. The star spot is accumulated to a light spot during exposure time, and the *i*th measurement star vector ${v}_{i}\in {R}^{3\times 1}$ in imaging frame $({X}_{S},{Y}_{S},{Z}_{S})$can be expressed using Eq. (1):

In celestial sphere frame, the reference star vector ${w}_{i}$ can be assured in the star catalog and the relationship between ${v}_{i}$ and ${w}_{i}$ is:

where $M\in {R}^{3\times 3}$ is the transfer matrix from celestial sphere frame to star sensor frame, as well as the attitude matrix of star sensor in celestial sphere frame. Based on two or more stars, $M$ can be calculated using TRIAD or QUEST algorithm [31, 32].The angle ${\gamma}_{ij}$ between two star vectors ${v}_{i}$ and ${v}_{j}$ is:

#### 2.2 A typical model

The pinhole model cannot fit with real star imaging exactly. With the effect of vibration during launch and time aging, the imaging system parameters may change. Different error models have been proposed [30, 33, 34]. Figure 2 shows a typical real measurement model for star sensors. In this article, we focus on parameter calibration and the difference between various models will not affect the parameter estimation using proposed method actually.

The relationship between true values and ideal ones is:

In this model, the measured star vector can be described as:

According to Eq. (1) and Eq. (3), the measured angle ${{\gamma}^{\prime}}_{ij}$ between star vectors ${{v}^{\prime}}_{i}$ and ${{v}^{\prime}}_{j}$ is:

Equation (6) can be also expressed as:

## 3. On-orbit calibration method

Calibration using interstar angle is an attitude independent method and is easy for operation. Based on interstar angles provided by star catalog and measured by star sensor, the neural networks and unscented Kalman filter (UKF) is combined: a simplified Back Propagation (BP) neural network is designed for coarse calibration; the UKF is adopted for precise calibration of all parameters. The calibration procedure is shown in Fig. 3.

#### 3.1 Coarse calibration with BP neural networks

BP neural network is a typical and widely used artificial neural network with characteristic of self-organization and self-learning [35–38]. Compared with Kalman filters, no initial information is needed to achieve the estimation with BP neural network. Imaging system calibration is a typical application area of neural networks and many researches have been done [39, 40]. Lyndon proposed a calibration method combining preliminary regression analysis and neural network, and the accuracy is better than traditional curve fitting [41]. Mark used the feed-forward neural network to achieve calibration. Instead of estimating parameters, the neural network implicitly learns the relationship between the camera and the global coordinate system [42]. It is proved to be effective but the structure is complicated.

In order to minimum the calculation time, only main point and focal length, which are the most important parameters for the optical system, are estimated firstly using BP neural network. The optical distortion is ignored and the ideal measurement model is adopted from Eq. (3).

According the input information and estimation parameter of coarse calibration, the input layer is the interstar angles ${w}_{i}^{T}{w}_{j}$, and the output layer is $Y={\left[f,{x}_{0},{y}_{0}\right]}^{T}$. In order to reduce the calculation time, the structure of neural network should be a simple one on the premise of the converged estimation results. In accordance with these analyses, a simplified BP structure is designed. Only one hidden layer is added in the structure, since it has been proved that neural network with one hidden layer can achieve mathematical approach to any rational function. The node number of hidden layer is 2N + 1, where N is the input star number. Thus, the structure for the coarse calibration is 1-5-3, as shown in Fig. 4.

The activation function of neural networks is the *Sigmoid* function:

The forward calculation is conducted to get the output *Y*:

The error is:

Define loss function *E* as:

The gradient-descent algorithm is adopted for the weights update. According the chain rule, $\frac{\partial E}{\partial W}$ is calculated.

For the output layer, $\frac{\partial E}{\partial W}$ is:

For $\frac{\partial {\widehat{v}}_{i}^{T}{\widehat{v}}_{j}}{\partial Y}$ in Eq. (12), it can be written as:

For the hidden layer, $\frac{\partial E}{\partial W}$ is:

The updated weights are ${W}_{mn}{}^{2}-\eta \frac{\partial E}{\partial {W}_{mn}{}^{2}}$ and ${W}_{mn}{}^{1}-\eta \frac{\partial E}{\partial {W}_{mn}{}^{1}}$ respectively, where $\eta $ is the learning rate of the neural network and $\eta =1$.

The loss function *E* is regarded as the ending signal. Considering the maximum interstar angle of a modern star tracker is smaller than several rads, the iterations stop when $E<\frac{1}{2}\times {0.0001}^{2}$. According to our investigation about modern star sensors parameters [8–11] and various simulations, the estimation error of the designed BP neural network for the star sensor can be less than 10%, which is a guarantee for the precise calibration with UKF afterwards. The details for estimation accuracy can be found in simulation and experiment.

With designed BP neural network, the rough values of focal length and main point are estimated without any priori information. In addition, since most cameras can be described using the pinhole model, the designed neural network structure can fit almost all star sensors, leading to a generalized coarse calibration approach.

However, the pinhole model does not fit the real measurement model exactly, especially when the optical distortion is ignored. Therefore the estimation results contain bias and further precise calibration for all parameters is needed.

#### 3.2 Precise calibration with UKF

Considering the nonlinear model of star vectors measurement described in Eq. (6), Unscented Kalman Filter (UKF) is adopted here for precise calibration [43–46]. With ensured initial guesses produced by BP neural networks, we can achieve precise estimation results and faster convergence speed through adopting the UKF.

The establishments of state and measurement equation are shown as followings:

Take the parameters as the state variables, and the system state equation is described as:

where ${X}_{k}$ contains all the estimated parameters at time*k*:${X}_{k}={[{x}_{0k},{y}_{0k},{f}_{k},{k}_{1k},{k}_{2k}]}^{T}$; ${\Phi}_{k|k-1}$ is the state transition matrix and in this calibration case ${\Phi}_{k|k-1}$. is $I$.

*I*is the identity matrix and its dimension is 5 here.

Supposing that the number of observed navigation stars at time *k* is *n*, the measurement equation is:

After the establishment of state equation and measurement equation, the calibration can be conducted using the UKF. The procedures of the precise calibration are:

- (2) Initialize the state parameter ${X}_{k}$:
where ${P}_{0}$ is the initial value of parameters variance.

- (3) Calculate the
*Sigma*point; - (4) Time update;
- (5) Measurement update.

The details of the UKF are not listed here and can be found in Wan’s descriptions [43]. Thus, all parameters, including focal length, main point and distortion can be estimated in the precise calibration.

## 4. Simulation and discussion

Several groups of simulation are designed for testification. The optical system parameters of our star sensor are listed in Table 1. The measurement model is adopted from Fig. 2 in Section 2. Since real star images are rarely downloaded from on-orbit star sensors, a digital platform is constructed to get calibration data, which is shown in Fig. 5(a). Several sequences of star images are captured randomly using Starry Night, which is a professional and powerful astronomical software. 10000 images are captured totally. To obtain navigation stars, the grid algorithm is adopted for star identification. To clearly show the performance of the calibration method, the quantity of picked navigation interstar angles in each frame is the same. Since the interstar angle quantity should be guaranteed to be larger than the number of estimated parameters, the number of stars is set as 4 and 5 interstar angles are picked randomly. Considering on-orbit environment, star location errors including are reference star vector error, imaging noise error, dynamic blurring error and star centroiding algorithm error are taken into account in our calibration environment. In section 4.1, 4.2 and 4.3, the star location error is set as 0 to exhibit the accuracy of the proposed method. In section 4.4, the star location error is added to show the robustness. The error chain for a single star measurement is described in Fig. 5(b). The initial value is: *f* = 0 mm and (*x*_{0}, *y*_{0}) = (500, 500) pixels, which are the worst guesses.

#### 4.1 Simulation only using EKF

The traditional extended Kalman filter (EKF) is reproduced firstly. The calibration results of focal length and main point are shown in Fig. 6. It is obvious that the results are divergent or convergent to a wrong value and the calibration is failed, let alone distortion calibration. The main reason is that the accuracy of EKF is affected by the initial values, which has been proved in Samaan’s method and Liu’s method [26, 29]. In our simulation, the EKF filter becomes invalid for the absence of proper initial guesses.

#### 4.2 Simulation only using BP neural network

Another calibration only based on BP neural network using all the 10000 star images is conducted. It can be seen from Fig. 7 that compared with traditional EKF, the estimation results can converge to correct values. The results of BP calibration after 10000 star images are: *f* = 43.2061 mm and (*x*_{0}*,y*_{0}) = (0.185,0.473) pixels. It can be deduced that with more sufficient star images, the calibration accuracy may be acceptable. However, the converging speed reduces rapidly and too many star images and calculation resource are needed for more accurate results, which is unacceptable for onboard system.

It can be concluded that both traditional EKF and BP neural networks cannot meet the desired calibration requirements individually.

#### 4.3 Simulation using proposed method

Calibration using proposed algorithm is conducted with no star location error. In the coarse calibration, the estimation of focal length and main point are shown in Fig. 8. After about 3700 star images, the estimation results are *f* = 42.18 mm, (*x*_{0}*,y*_{0}) = (1.552,-0.450) pixels. It can be seen that a rough estimation can be achieved using designed BP neural network. However, since the distortion is ignored as discussed in section 3.1, the distortion parameters cannot be estimated. The un-calibrated distortion also results in bias of the coarse calibration results.

With initial information adopted from coarse calibration, the precise calibration is conducted and the results are shown in Fig. 9. The initial distortion parameter is set as 0. After about 60 star images, the estimation is accomplished. The estimation results are *f =* 43.18157mm, (*x*_{0}*,y*_{0}) = (0.0154, −0.0037) pixels, the distortion parameters are *k*_{1} = −5.567e-6, *k*_{2} = 4.553e-7. The relative error can reach (43.18157-43.18156)/43.18156≈0.000023% for the focal length, which is extremely low. The relative error of main point is (0.0154, −0.0037)/(500, 500)≈(0.0031%, 0.00074%). The relative errors of distortion are: *k*_{1} is 1.86% and *k*_{2} is 0.77%. All the parameters are estimated precisely. The calibration results are listed in Table 2.

The distortion error on the focal plan is shown in Fig. 10. For the original distortion in Fig. 10(a), the star location error can be about 1.5 pixels at the edge of the FOV, which brings inaccuracy for attitude measurement. After calibration in Fig. 10(b), the star location error is less than 0.005 pixels, which indicates that the distortion parameters are estimated successfully using proposed method. Besides, for other distortion models, such as distortion with both radial and tangential errors or other irregular distortions, the parameter estimation can also be achieved using proposed method with sufficient interstar angles.

Considering the resource needed for calibration, about 3700 and 80 star images are needed in the coarse and precise calibration respectively. Less than 4000 star images are needed totally to achieve satisfied calibration results.

Considering the time consumption, our calibration can be achieved provided that the calibration data is downloaded to the ground station or the calibration is conducted on an individual chip which is parallel with system navigation chips. For calibration using navigation chip, the calibration can be conducted after the navigation calculation in the time interval. So long as one single calibration iteration can be finished within one time interval, the calibration can be also accomplished with sufficient time.

#### 4.4 Robustness with respect to star location error

As the original information of calibration, the star location error will undoubtedly affect the calibration results. Several other calibration tests with different noise level using proposed algorithm are conducted. A Gaussian noise with a mean of 0 pixels and standard deviation of 0, 0.05, 0.1 and 0.2 pixels are added in star location error in each calibration case. Take star location error = 0.05 pixels for example, reference star vector error is set as 0.0001 pixels, imaging noise error is set as 0.04 pixels, dynamic blurring error is set as 0.03 pixels and star centroiding algorithm error is set as 0.01 pixels, then the star location error is $\sqrt{0.\text{0}{001}^{2}+0.\text{0}{4}^{2}+{0.03}^{2}+{0.01}^{2}}\approx 0.05$ pixels. The star location error can be changed through adjusting the imaging noise and dynamic blurring error. The calibration results including Root Mean Square Error (RMSE) are listed in Table 3.

It can be seen that the estimated mean values are slightly influenced even with loud noise. The relative errors are at the same level comparing ideal conditions and loud noise for all parameters. Especially for focal length and main point, the estimation error is almost not influenced, which is a guarantee for the calibration results.

However, the RMSE is greatly influenced by star location error. For focal length, given that the star location error is 0, the RMSE is about 1.67e-6, showing the capability of proposed method. When the star location error is 0.2 pixels, the RMSE is 4.12e-5, which is about 20~30 times larger. The RMSE of main point is more seriously influenced by star location error. When the star location error is 0.2 pixels, the RMSE is about 100 times larger than ideal conditions. Therefore, the calibration results, especially the RMSE results, can show the level of noise, proving the validity of proposed method. The calibration method is proved to be effective and reliable under different noise levels.

It can be concluded from various simulations that, with proposed calibration method, all parameters can be estimated without any priori information, which is beyond the capability of traditional EKF method. Accurate calibration results are achieved. The star images needed in our self-initialization calibration is less than traditional BP neural networks. The proposed method is also proved to have a good robustness with respect to star location error.

## 5. Experiment

Ground experiments are conducted for further verification. A turntable, a single star simulator and a star sensor with wide FOV are included in the experiment. The parameters of facilities are listed in Table 4. The placement of the turntable and the star simulator is calibrated using a theodolite in advance, as well as the installation between the turntable and the star sensor. The experiment setup is shown in Fig. 11. The star vector is obtained at different turntable orientation. The star points are uniformly distributed over the range of (−7°, + 7°) × (−7°, + 7°) with a span of 0.5°, thus generating a matrix with 15 × 15 star spots. The locations of star spot on focal plan are shown in Table 5. 5 measured interstar angles are randomly picked for calibration in each frame.

The calibration results and the statistical characteristics are listed in Table 6. Parameters based on traditional laboratory calibration are regarded as the reference. In the experiment, the parameter values are corresponding with that of laboratory calibration, showing the validity of our method. The number of needed star images is at the same level with that in simulation, proving the accordance between simulation and experiment.

The calibration accuracy is lower than that of laboratory calibration. For focal length, the absolute error is 0.0008 mm and the relative error is 1.59e-3%, which is larger than that in simulation. The main point error is less than 0.05 pixels and the relative error is less than 1.45e-2%. What’s more, the RMSE is larger than that in simulation significantly, especially for the main point.

The main reason for the increased noise of the calibration results is that the performance of facilities and various disturbances during ground experiments are inevitable, bringing in uncertain errors. These values are at the same level of noise introduced in by facilities. In addition, this situation may be ameliorated in real on-orbit environment with accurate star catalog and less vibration and other interferences.

It can be concluded from the experiment results that, almost the same estimation results can be achieved using the self-initialization on-orbit calibration method comparing with laboratory calibration, demonstrating our superiority in terms of validity and accuracy.

## 6. Conclusions

On-orbit calibration is an essential guarantee for star sensor operation accuracy and plays an important role in spacecraft GNC system. Most existing on-orbit calibration need initial guesses and priori information, which may lead to declined star sensor measurement accuracy or even failed calibration. In this paper, a self-initialization on-orbit calibration method for star sensor is designed combining BP neural networks and UKF. Ground data or preset parameters are not required in our calibration. Only with measured interstar angles, star sensor on-orbit calibration can be achieved and all parameters can be estimated. The on-orbit calibration accuracy is extremely high, especially for focal length and main point. Calibration results in simulation and experiment demonstrate our superiority in terms of convenience, autonomy, accuracy and robustness. The proposed method has the potential to be a universal calibration method for most on-orbit star sensors.

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