Closed-Form Thrust Acceleration for Optimal Power-Limited Trajectories
Abstract
A simple class of minimum-fuel power-limited trajectories has recently been
discovered. The optimal thrust acceleration can be expressed in closed-form
as a simple, time- varying scalar multiple of the velocity vector. The thrust
direction is then always aligned with the velocity vector. This thrust law
satisfies the first-order necessary conditions for an optimal trajectory
in an arbitrary gravitational field. The optimal trajectory can be calculated
using only the state equations, without simultaneous calculation of the
primer vector.
These solutions form a subset of optimal power-limited trajectories, and
have some limitations on their use. No plane change can be provided by the
thrust law considered. Trajectories which have an unspecified final velocity,
such as an optimal intercept, cannot be accomplished with this thrust law.
However, optimal rendezvous and orbit transfer can be realized by these
trajectories.
The problem of maximizing the change in total energy is examined. It is
shown that thrusting along the velocity vector cannot satisfy the necessary
conditions for an optimal trajectory, even though the terminal conditions
dictate that the final thrust must be along the final velocity. This result
is consistent with numerical results showing that optimal energy trajectories
do not simply thrust along the velocity vector, even though this thrust
law maximizes the instantaneous rate of change of the total energy.