Lissajous Trajectory Research Articles

Station-keeping method for libration point trajectories

Three-dimensional orbits in the vicinity of the interior libration point (Li) of the Sun-Earth/Moon barycenter system are currently being considered for use with a number of missions planned for the 1990s. Since such libration point trajectories are, in general, unstable, spacecraft moving on these paths must use some form of trajectory control to remain close to their nominal orbit. The primary goal of this effort is the development of a Stationkeeping strategy applicable to such trajectories. A method is presented that uses maneuvers executed (impulsively) at discrete time intervals. The analysis includes some investigation of a number of the problem parameters that affect the overall maneuver costs. Simulations are designed to provide representative station- keeping costs for a spacecraft moving in a libration point trajectory, and preliminary results are summarized. RAJECTORY planning for a number of scientific mis- sions scheduled for launch in the 1990s includes the possi- ble use of three-dimensi onal halo or Lissajous orbits in the vicinity of the interior L! libration point of the Sun-Earth/ Moon barycenter system.1 This effort is directed toward the development of a Stationkeeping strategy that can be used to maintain spacecraft near such nominal libration point trajec- tories. A significant number of analyses have been completed that involve Stationkeeping methods for Earth orbiting satel- lites; maintaining a spacecraft on a libration point orbit, how- ever, has received limited attention. In the late 1960s, Far- quhar2 developed a number of possible Stationkeeping strat- egies for libration point orbits. Later, in 1974, a station- keeping method for spacecraft moving on halo orbits in the vicinity of the Earth-Moon translunar libration point L2 was published by Breakwell et al. 3 In contrast, specific mission requirements influenced the design of the Stationkeeping strat- egy for the first libration point mission. When the Interna- tional Sun-Earth Explorer-3 (ISEE-3) satellite was injected into a halo orbit associated with the interior L! libration point of the Sun-Earth system in 1978, a series of maneuvers exe- cuted at approximately three-month intervals was used for Stationkeeping.4 More recently, a series of papers have pre- sented results from studies that use Floquet and invariant manifold theories to develop a ''loose Stationkeeping strat- egy for halo-type orbits.58 Similar to ISEE-3, the method also uses discrete maneuvers, applied at varying time intervals, that control the trajectory near the nominal path. A significant goal of this study is the development of a potentially ''tight control strategy for Stationkeeping that can be applied to both halo and Lissajous trajectories (as well as other possible types of libration point paths). In this approach, the allowable deviation of the actual trajectory relative to the nominal path can be varied over a wide range depending on mission require- ments. Of course, low costs are desirable as well.

Three-dimensional human pattern visual evoked potentials. I. Normal subjects

Pattern visual evoked potentials (PVEPs) were obtained from 30 normal adultvolunteers, recording from both a conventional horizontal occipital array and three orthogonal bipolar antipodal channels approximating the three dimensions of space. Central and eccentric fixation of 60′ checks and central fixation of 30′ checks under binocular and monocular viewing conditions was employed. The three antipodal wave forms were displayed as a single 3-D Lissajous trajectory which contained four apices, corresponding to P40 (apex A), N70 (apex B), P100 (apex C) and N125 (apex D). The 3-D evoked potentials depicted the dynamic nature of the human PVEP in terms of changes in the 3-D voltage-voltage-voltage plots of the recordings. The orientation of the A-B, B-C and C-D curvilinear segments reflected the stimulating condition (central fixation vs. right vs. left hemi-field stimulation) for all subjects with more accuracy than did the wave forms from the conventional array. Spherical statistical methods are described for quantifying and evaluating 3-D evoked potential recordings.

Three-dimensional human pattern visual evoked potentials. II. Multiple sclerosis patients

Pattern visual evoked potentials were obtained from 46 patients with definite relapsing/remitting multiple sclerosis, using both a conventional 5-channel occipital array and a 3-D recording technique consisting of three bipolar derivations approximating the three dimensions of space. These three orthogonal wave forms were displayed as a 3-D Lissajous trajectory for each subject. Two of the 15 patients with completely normal conventional pattern VEPs had abnormalities of the orientation of the B-C curvilinear segment of the 3-D pattern VEPs. Delays in the first major occipital positive component (P100) were evident using both techniques; the correlation between P100 latency and the latency of the corresponding trajectory apex was r = 0.99 ( P < 0.01). Post-chiasmal MRI abnormalities were associated with 3-D VEP orientation abnormalities. Three-dimensional pattern VEPs are moderately more sensitive than conventional pattern VEPs at detecting dysfunction posterior to the optic chiasm in demyelinating disease and do not require the use of eccentric fixation to do so.

Numerical determination of Lissajous trajectories in the restricted three-body problem

In previous studies, Lissajous trajectories associated with the collinear libration points in the restricted three-body problem have been successfully computed analytically to at least third-order. Those approximations are utilized to determine such trajectories numerically for an arbitrary, predetermined number of revolutions in the rotating frame, for the case of circular primary motion. The numerical approach first identifies target positions at specified intervals along the trajectory and locates a continuous path through those points with velocity discontinuities. Then the Δv are simultaneously reduced in an iterative process. Such trajectories have been constructed in various primary systems, for a wide range of orbit sizes and a large number of revolutions.

Application of the Three-Channel Lissajous Trajectory of Auditory Brainstem-Evoked Potentials to the Question of Generators1: Application des trajectories de Lissajous à trios canaux des potentials évoqués auditifs à l'étude des générateurs

In studies of generators of auditory brainstem-evoked potentials (ABEP), depth recordings and clinical correlates may not reflect all the activity involved. Although the 'inverse problem' is mathematically ill-posed, a synthesis of physico-anatomical data with a three-channel Lissajous trajectory (3CLT) may indicate the actual number and identity of generators. A 3CLT of ABEP in three-dimensional voltage space, is isomorphic with the vector tip representing the momentary centrally located dipole in a uniform, spherical volume conductor. This isomorphism enables approximations of ABEP dipole equivalents to be made. 3CLT summarizes the surface activity in a single plot. It accentuates correlations within the orthogonal records, such as planarity of segments, allowing segmentation of the analysis epoch into components. These properties of 3CLT simplify the analysis of parametric effects on the surface distribution of ABEP components. The results of applying this technique indicate the experimental conditions under which the assumptions for the approximation of generators are closest to being fulfilled. Of all the points along the trajectory, apices (points with maximum absolute amplitude and curvature) have the most unique generators. The 3CLT to low-intensity and high-rate clicks, indicate more unique generators than the customarily used high-intensity slow-rate clicks, in which considerable temporal overlap of generator activity exists.

Field analysis of auditory evoked brainstem potentials

Surface distribution maps of auditory evoked brainstem potentials (AEBPs) are presented for time instants corresponding to the waves of major interest (I, III and V) in humans. Maps are shown to be fully consistent with the results of two approaches that have been recently proposed for the analysis of AEBPs, namely dipole localization methods and three-channel Lissajous trajectories, thus confirming the technical feasibility of methods for extracting information on the sources of AEBPs from surface data.