Today, mechanical tracking systems are becoming increasingly compact, enabling a new range of civil and military applications. These include aerial laser scanning, for which Risley prisms are used. In Risley systems, the so-called inverse problem, which focuses on obtaining the angles of the prisms for a given target coordinate, has not yet been solved mathematically. As a consequence, approximate approaches have been used, but the solutions obtained have significant errors and a lack of precision. To improve accuracy, iterative methods, which are computationally intensive, have also been implemented. In this paper, an analytical process which we call the geometric method is presented, and we verified that this strategy highly improves accuracy and computational speed. Using this method in an iterative process gives accuracies of up to 1 pm in only three iterations. This high accuracy would allow the geometric method to be applied in fields such as lithography, stereolithography, or 3D printing.

Today, mechanical tracking systems have been downsized to allow them to be used in the field of airborne laser communications and in the military domain. Risley systems are used for this purpose, which work by directing a beam of light to a given target point, this procedure is commonly known as the inverse problem. In this paper, an analytical method, the geometric method, has been designed and developed to determine the beam steering in a Risley system and solve the inverse problem. The method focuses on different geometric shapes, like circumference or ellipse, that are described when the beam passes through the second prism. The accuracy and efficiency of the geometric method has been analysed and found to be faster than the two-step method. Furthermore, the geometric method has been implemented in an iterative process and an accuracy of up to 1 pm has been achieved. This high accuracy would allow the geometric method to be applied in fields such as lithography, stereolithography or 3D printers.