## 5 Initial Orbit Determination Methods
Building upon this, the method utilizes observational geometry to ascertain the position and acceleration of the target, enabling the determination of the target's six orbital elements. Theoretical research and application instances show that the Laplace method proves to be a more straightforward and practical method for orbit determination missions, particularly when dealing with orbit determination problems with very limited observational data [58]. However, it is crucial to acknowledge the inherent difficulty in accurately determining the geocentric distance of space targets due to errors in determining the target's geocentric distance relative to angle information from the observation platform. This challenge becomes particularly pronounced in scenarios with sparse observational data during short-arc orbit determination. The inherent limitations of the Laplace method make it difficult to precisely determine the geocentric distance of observed targets, potentially leading to an ill-condition during the matrix inversion process when calculating the position vector of the target, sometimes even resulting in orbit determination failure [43; 59; 60]. Distinct from the Gauss method and the Laplace method, the Gooding method and the double r method are other two widely recognized techniques for initial orbit  
Figure 5: Schematic diagram of angle-only orbit determination based on three observations. determination [61; 62]. [PERSON] systematically compared the Gauss method, the Laplace method, the double r method and the Gooding method in terms of accuracy, convergence, robustness and suitability for space-based platforms. The results consistently showed that the Gooding method exhibited optimal performance in initial orbit determination problems in the majority of cases [63]. When estimating the direction of the target, the accuracy of the Gooding method typically surpassed other methods by several orders of magnitude. In terms of the shape estimation of orbits, the precision of the Gooding method and double r method was generally comparable. In scenarios involving near-polar orbits and issues where the initial distance estimation for the target orbit is unknown, the Laplace method often performs the best. To further enhance the precision and speed for solving the initial orbit determination problems, [PERSON] et al. investigated the observability of the target relative to the monitoring platform. They proposed an algorithm for determining the shape of the target orbit under the condition of angle-only measurements [64]. [PERSON] and others addressed observability issues in the angle-only initial orbit determination problems through nonlinear second-order Volterra sequences [65; 66]. Subsequently, [PERSON] et al. optimized the algorithms proposed by [PERSON] and achieved higher computational efficiency [67]. [PERSON] et al. developed another orbit determination method by setting the camera offset from the vehicle [68; 69]. [PERSON] et al. employed neural networks to map the observation vectors to space orbits, effectively addressing the nonlinear challenges introduced by complex dynamic models [70]. While machine learning methods exhibit commendable performance in orbit determination, they concurrently escalate the computational demands for space situational awareness tasks, significantly hampering the operational efficiency of associated algorithms. Within the highly constrained computational hardware of onboard computers, these substantial increases in computational workload may even lead to a considerable escalation in overall computation. According to the research of [PERSON] et al. [71], GPUs can extend the computing speed of CPU-based algorithms, providing the possibility of real-time computing. [PERSON] et al. [72] pointed out that GPU can improve the computational efficiency by concurrently executing the matrix operation process that decouples each other in the calculation of orbit dynamics problems, thereby greatly reducing the computational time complexity.