Zero miss distance of 3D PPNG with time-varying speed

Using the exact nonlinear kinematics, the zero miss distance (ZMD) property of the three dimensional (3D) pure proportional navigation guidance (PPNG) law is examined for a missile with a time-varying speed to intercept an arbitrarily maneuvering target in the space. It is proven that, a navigation constant larger than 1 is the only condition required to guarantee the ZMD property of 3D PPNG, provided that 1) the missile is launched toward the target, and 2) the missile has a speed superiority over the target during flight. To this end, a new description of the 3D engagement geometry is proposed, in which the concept of plane angle is introduced to describe the cross-coupling in the nonlinear kinematics. Based on this, the approach for analyzing the ZMD property of PPNG in planar pursuit situations is extended successfully to general 3D cases: firstly, a certain time sequence and an asymptotic time function dependent on the maximum speed ratio are defined, based on which the upper bound of the relative range between the missile and the target is assured; then, it is proven by contradiction that there exists a subsequence such that the relative range is asymptotically converging to zero. Our methodology is expected to be useful for the design and analysis of other 3D missile guidance laws.