A simple control law for low-thrust orbit transfers
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Engineering topics
Publications and source records attributed to Petropoulos, A. E..
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Interplanetary missions which use low-thrust, high specific impulse propulsion can further capitalise on the capabilities of the propulsion system by using it to effect escape from the launch body or capture at a target body.
With the complexities in computing optimal low thrust trajectories, easily-computed, good sub-optimal trajectories provide both a practical alternative for mission designers and a starting point for optimisation. The present paper collects in one place for easy reference and comparison several exact solutions that have been obtained in the literature over the last few decades: the logarithmic spiral, Pinkham's variant thereof, Forbes spiral, the exponential sinusoid, the case of constant radial thrust, Markopoulos's Keplerian thrust arcs, Lawden's spiral, and the analogous Bishop and Azimov spiral.
Several methods for determining the characteristics of low-thrust transfers around minor planets are examined and compared.
Ten male subjects underwent two types of adaptation paradigm designed either to enhance or to attenuate the gain of the canal-ocular reflex (COR), before undergoing otolith-ocular reflex (OOR) testing with constant velocity, earth horizontal axis and pitch rotation. The adaptation paradigm paired a 0.2 Hz sinusoidal rotation about an earth vertical axis with a 0.2 Hz optokinetic stimulus that was deliberately mismatched in peak velocity or phase and was designed to produce short-term changes in the COR. Preadaptation and postadaptation OOR tests occurred at a constant velocity of 60 degrees/sec in the dark and produced a modulation component of the slow phase velocity with a frequency of 0.16 Hz due to otolithic stimulation by the sinusoidally changing gravity vector. Of the seven subjects who showed enhancement of the COR gain, six also showed enhancement of the OOR modulation component. Of the seven subjects who showed attenuation of the COR gain, five also showed attenuation of the OOR modulation component. The probability that these two cross-axis adaptation effects would occur by chance is less than 0.02. This suggests that visual-vestibular conditioning of the yaw axis COR also induced changes in the pitch axis OOR. We thus postulate that the central nervous system pathways that process horizontal canal yaw stimuli have elements in common with those processing otolithic stimuli about the pitch axis.
The vertical eye movements in humans produced in response to head-over-heels constant velocity pitch rotation about a horizontal axis resemble those from other species. At 60 degrees/s these are persistent and tend to have non-reversing slow components that are compensatory to the direction of rotation. In most, but not all subjects, the slow component velocity was well characterized by a rapid build-up followed by an exponential decay to a non-zero baseline. Super-imposed was a cyclic or modulation component whose frequency corresponded to the time for one revolution and whose maximum amplitude occurred during a specific head orientation. All response components (exponential decay, baseline and modulation) were larger during pitch backward compared to pitch forward runs. Decay time constants were shorter during the backward runs, thus, unlike left to right yaw axis rotation, pitch responses display significant asymmetries between paired forward and backward runs.