Optimization of Midcourse Velocity Corrections

Abstract

The concept of a six-dimensional state space is used to develop the fundamental equations of linearized midcourse guidance. Both fixed end-point (fixed-time-of-arrival) and variable end-point (variable-time-of-arrival) problems are considered. It is shown that the variable-time-of-arrival problem can be simplified mathematically by the introduction of a special coordinate system, which is called the critical-plane coordinate system.A method is developed for determining the optimum time at which to apply a single midcourse correction the effect of which is to satisfy a set of position constraints. The correction is “optimum” in the sense that its magnitude is minimized. For a given nominal trajectory, the time of the correction depends on the predicted miss vector at the destination. The method is particularly simple to apply in the case of variable-time-of-arrival guidance; by exploiting the critical-plane coordinate system, a single curve can be prepared prior to the flight to indicate the optimum correction time as a function of a miss parameter which is determined from in-flight navigational measurements.Multiple-correction strategies are then investigated. A method is developed for determining an optimum schedule of midcourse corrections. The optimum schedule is the one for which the sum of the magnitudes of all corrections is minimized. It is proved that the number of corrections in an optimum schedule is no greater than the number of constraints to be satisfied at the nominal time of arrival at the destination.In position-constrained variable-time-of-arrival guidance there are only two constraints at the nominal time of arrival; hence there are at most two corrections in the optimum schedule. The optimum two-correction strategy is compared with the optimum single-correction strategy. It is shown that for certain ranges of the miss parameter two corrections can effect a saving in total magnitude of velocity correction, while in other ranges no improvement can be obtained from two corrections. A geometric construction, based on the theory of convex sets, is used to determine the ranges of miss parameter in which two corrections are preferable, and also the times and components of both corrections when they are preferable.It may be noted that the developments in this paper are deterministic rather than statistical. No consideration is given to the uncertainties of the navigational measurements; it is assumed that a sufficient number of measurements has been made during the flight so that the error in the predicted miss vector at the destination is negligible. The control action taken is determined by the predicted miss vector.KeywordsOptimum ScheduleReference TrajectoryCorrective EffectVelocity CorrectionCritical PlaneThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.