Linear guidance laws for space missions

Abstract

A general analysis of guidance laws is presented along with a review of the propagation of state perturbations. Linearized guidance laws are derived for fixed and variable time of arrival, and for the Shuttle phasing and height guidance maneuvers. In general, six interrelated partials are involved in each guidance law. These partials relate the maneuver delta velocity and time of flight to perturbations in initial position, velocity, and time. The partials can be expressed as a function of a single fundamental guidance partial, which relates the maneuver to the final position perturbation, and the partials from the upper row of the augmented time transition matrix (augmented with the dynamical state vector). Guidance can be interpreted as a form of offset targeting. PACE trajectories involve a series of coast phases separated by maneuvers. The maneuvers can be derived by imposing sufficient guidance constraints to uniquely define the maneuver. The guidance constraints can be imposed on the maneuver, postmaneuyer trajectory, or some combination thereof. Impulsive maneuvers may be assumed for many space missions. The impulsive maneuvers can be computed exactly or with a linearized model. The linearized model could be used for real-time software because trajectory perturbations are generally small, but a more likely use is in preflight covariance analysis. The linear model contains guidance matrices for up- dating through the maneuvers, and transition matrices for up- dating through the coasting phases. This paper presents two general techniques for solving the linear guidance problem. Most applications of linearized guidance laws have been based on fixed- and variable-time-of-arrival guidance.1'4 Many other linearized laws can be introduced by imposing dif- ferent constraints on the maneuver and postmaneuver trajec- tory.5'6 Some general properties of linearized guidance laws were established by Cicolani.7 A general solution to the linear guidance problem was proposed by Tempelman5 by introduc- ing a generalized linear constraint equation. This paper at- tempts to define some general techniques for establishing the relationship between the maneuver and the perturbations in position, velocity, and time using a more intuitive approach than presented in Ref. 5. Fixed- and variable-time-of-arrival guidance laws are reviewed to demonstrate the techniques and to serve as an introduction to more complicated maneuvers, such as the Shuttle phasing and height maneuvers.