Missile Guidance by Three-Dimensional Proportional Navigation

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Abstract
Following a brief comparison of three collision-seeking types of navigation—pure pursuit, constant-bearing collision, and proportional navigation—the usual definition of planar proportional navigation is extended to three dimensions. Based upon a simple criterion for optimum navigation to the line-of-sight motion, a proper formulation is found in terms of the geodesic and normal curvatures of the missile path on the surface generated by the line of sight. By a suitable choice of a reference coordinate system, the missile-target kinematic relationships are then linearized, assuming relatively small deviations of the missile from a collision course. Upon combining these ``geometry feedback'' equations with the equations of guidance, the missile trajectory is found to be given in terms of two independent linear differential equations of identical form and of one higher order than the missile transfer function. Typical solutions are found for some simple cases. The character of the trajectory is shown to depend on an ``effective navigation constant'' proportional to the missile navigation constant (or gain) and the ratio of missile speed to closing speed; a value of this parameter greater than two is found to be necessary to insure finite terminal missile acceleration. Two example trajectories are calculated from both the exact and the linearized trajectory equations to indicate the accuracy of the linearization.

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