Theoretical minimum RMS error of a Deployable Mesh Reflector

Communications, scientific imaging, and aerospace satellite missions demand antennas with large and precise apertures. In addition, several aerospace applications require the use of competitive designs for large reflectors with high performance that must be cheap to produce, fast to deploy, and reliable in their functioning. In recent decades, Deployable Mesh Reflectors (DMR) have attracted the attention of several aerospace companies due to their wide applications. The key features that characterize the geometry of DMR systems are closely connected with volume constraints of launch vehicles, mainly because of budget problems [1]. To maintain excellent reflective qualities and meet the prescribed bandwidth requirements, the reflector surface must be as close as possible to the shape of a paraboloid. Most of the methods used in the literature define the best surface of the reflector as the one passing through the nodes of the cable system of the front net [2]. In this case, the RMS error depends on the distance between the nodes of the front net with respect to the desired working surface. In this work, however, we want to focus on the amount of energy that hits the feed, thus investigating the best topology of the net that guarantees a greater concentration of the incident rays directed towards the focus of the paraboloid. At the same time, it is important to examine what is the minimum ideal value that can be achieved with an assigned geometric configuration of the cable net. In this work, the best-fit paraboloid that guarantees the best energy contribution to the feed can be achieved thanks to the formulation of an optimization algorithm which, by varying the position of the nodes of the cable net, minimizes the distance of each reflected ray with respect to the focus of the paraboloid. To demonstrate the validity of the proposed method, the best topology of an offset-feed reflector is proposed. The set of points located in the left part of the Figure 1 represents the points of minimum distance between the reflected rays coming from the center and from the vertices of each triangular facet of the cable net with respect to the focus. These points are located in a very small volume near the feed, thus demonstrating the goodness of the algorithm. Since the RMS measurement method consists in calculating the average value between the points of minimum distance of each reflected ray with respect to the focus of the paraboloid, we consider to examine a single triangle of the net and evaluate its average value. In mathematical terms, this is equivalent to the integral of the distance function between every point inside of it with respect to the center, divided by its area. In this way we are able to evaluate the effectiveness of the optimization algorithm described above by measuring the difference between the real RMS error and the ideal one.