3D Unified Approach to Missile Guidance Laws

Since the proportional navigation guidance law was first introduced, many of the researchers had proposed different methodologies to investigate the corresponding performances of all the existing guidance laws. In 1997, Yang and Yang introduced an unified approach, in that paper the authors found that under the proposed framework, all the existing guidance laws, namely IPN, TPN and PPN, are indeed special cases of the mentioned general guidance law. However, their results were restricted to two dimensional space. In this paper, the author not only extends the results to three dimensional space, but also to GIPN, GTPN and PPN. Unlike conventional researchers, a modified polar coordinate (MPC) is adopted. It is shown that with the property of this modified polar coordinate, for the line of sight based guidance laws (GIPN and GTPN) the number of differential equations required to fully describe the relative dynamics can be reduced from six to three, however, for the missile's velocity based guidance law, i.e. PPN, five differential equations are required. In the mean time all the terms of differential equations involve only products and additions of variables. For all the mentioned guidance laws in this paper, only two transformed variables are required to describe the capture region, while the third variable is required to provide the condition of finite turn rate. For each type of guidance laws (GIPN, GTPN and PPN), the corresponding time-to-go's are also derived.