Lens distortion is a common problem in the optical system of star sensors. Vibrations during launch, changes in the space thermal environment, and other factors may cause changes in the optical system and lead to lens distortion. The changes in the optical system include the refractive index changes of optical materials, the thickness, curvature radius, and surface shapes of optical lenses, and the distance changes between optical elements. These distortions can change the focal length and principal point and can cause various nonlinear distortions of the optical system, leading to errors in the angle information obtained in the star sensor, thereby affecting the accuracy of navigation and attitude control. These errors may accumulate in long-duration missions and lead to poor system performance. Therefore, eliminating lens distortion and improving the measurement accuracy of star sensors become necessary for maintaining the attitude output accuracy of star sensors.

In response to the accumulation of lens distortion in star sensors due to mechanical vibrations, temperature changes, radiation, and solar radiation pressure during launch, as well as poor attitude accuracy caused by the optical distortion of uncalibrated low-cost cameras, we creatively introduce the non-parameterized B-spline algorithm into the widely used model-based correction algorithm. This approach decouples lens distortion into parameterized coarse calibration and non-parameterized fine calibration. The main advantage of B-spline curves is their flexibility and local controllability. Compared to the traditional interpolation methods, the shape of B-spline curves can be controlled locally by adjusting the positions of control points without affecting the entire curve. In the coarse calibration stage, the Levenberg-Marquardt algorithm is introduced to optimize the principal point and focal length, and the results are used as parameters for data preprocessing. Additionally, the right ascension, declination, and rotation angle of each frame image are also used as inputs. After the data preprocessing, the pixel coordinates of each star point on the image plane are produced. Lastly, a multi-layered structure of bicubic B-spline grids is constructed, achieving sound correction of global distortion and addressing local nonlinear distortions. This approach reduces the requirements for lens manufacturing and improves the attitude accuracy of on-orbit star sensors.

We conduct a simulation to simulate various distortion situations that may occur (Table 2), determine the optimal number of layers for the B-spline grid (Fig. 5), and verify the compensation ability of the non-parameterized B-spline algorithm for distortion (Fig. 6). In terms of distortion correction and time consumption, comparisons are made with the BP network algorithm optimized by genetic algorithm and neural network algorithm. Simulation experiments show that B-splines can effectively handle various distortions of star sensor lenses with theoretically high accuracy (Table 3). Compared with the neural network algorithm and genetic algorithm, our algorithm achieves attitude output accuracy at the sub-arcsecond level after correction, which is an order of magnitude better than the arcsecond level accuracy of the neural network algorithm and genetic algorithm (Table 4). To verify the on-orbit feasibility of the algorithm, 850 images transmitted from a star sensor of a remote-sensing satellite in the sun-synchronous orbit are selected for calibration testing. After the calibration with this algorithm, the position deviation of star points is reduced from 0.55766 to 0.23706 pixel (Table 6), and the measurement accuracy is improved from 5.857″ to 2.775″, demonstrating high feasibility. Compared with common ground calibration algorithms, this algorithm shows higher calibration accuracy and requires only a few hundred milliseconds to train all data.

This paper proposes a rapid spline-based on-orbit self-calibration method, which decouples distortion into parameterized coarse calibration and non-parameterized fine calibration. Through the design of the B-spline grid and the self-calibration algorithm, fast and accurate correction of star-sensor distortion is achieved. Through simulation and on-orbit experiments, the effectiveness and robustness of the method are proved. This research provides an effective method to improve attitude accuracy in on-orbit self-calibration of star sensors. It also provides theoretical and experimental foundations for the development of cost-effective star sensors.