Local Horizontal Plane Research Articles

A Method for Rapid Measurement of the Deformation and Spatial Attitude of Large Radar Antennas Inside Radomes

It is challenging to measure, analyze, and control the deformation and space attitude of antennas in narrow radomes through geometric measurement to ensure the accuracy of phased array radar application. In this manuscript, a method for rapid measurement of antennas by adopting the combination of photogrammetry system, laser tracker and total station was proposed. The photogrammetric method was used with a homemade auxiliary tool to measure and calculate the antenna’s deformation under different temperatures and pitch angles. Mapping relations between the instrumental coordinate systems of photogrammetry system, laser tracker and total station were established through common points, including target balls and reflectors. Coordinates of mark points on antenna surface were converted from photogrammetric coordinate system to laser tracker’s and total station’s coordinate system. Thus, antenna’s pitch angles relative to the local horizontal plane and deflection angles relative to the north of the engineering coordinate system were calculated. Planar fitting, coordinate transformation and space attitude calculation were all carried out in the Spatial Analyzer (SA) software. The photogrammetric method detected the antenna’s gravitational deformation sensitively, the maximum Root Mean Square Residual (RMSR) of the reference length was $54~\mu \text{m}$ , and that of the mark points was $59~\mu \text{m}$ , which was stably in line with its nominal accuracy. Deviations caused by coordinate transformation had no significant effect on the calculation of antenna’s spatial attitude, for the maximum deviation between the converted and measured coordinates in laser tracker’s coordinate system was 0.139 mm, and that in total station’s coordinate system was 1.037 mm, both of which were within reasonable limits of the derived theoretical maximum deviations between different instruments. As a result, the maximum deviation between the calculated value and servo system’s nominal value of antenna’s pitch angle was $59.4''$ , and that of antenna’s deflection angle was $91.87''$ . This method’s efficiency was greatly improved by about ten times compared with traditional methods through statistics and estimates.

The Foucault pendulum and the additivity of infinitesimal rotations

The purpose of this work is to present the Foucault pendulum precession without making recourse to the Coriolis acceleration or to geometrical methods. The exposition makes use of the fact that, contrary to finite rotations, infinitesimal rotations about different axes in three dimensions are both additive and commutative and that these important properties are readily extended to their time-derivatives, i.e. angular velocities. The angular velocity of the Earth about the polar axis is then decomposed into a pair of orthogonal angular velocity vectors having a common start point at the center of the Earth; the first vector is parallel to the axis that, in the local horizontal plane, points toward the north, while the second vector is directed upward, pointing toward the site where the pendulum is positioned. For the validity of the procedure it is essential to show that an infinitesimal rotation about the horizontal axis can be shown to have a negligible effect on the orientation of the plane of oscillation of the pendulum. At this point, only an infinitesimal rotation about the vertical axis is enough for calculating the change of orientation of the pendulum. Finally, owing to the constancy of the Earth’s spin, the obtained result can be readily extended to any finite rotation of the Earth, for example through the angle 2π that corresponds to a whole sidereal day.

The evaluation of unmanned aerial system-based photogrammetry and terrestrial laser scanning to generate DEMs of agricultural watersheds

Agricultural watersheds tend to be places of intensive farming activities that permanently modify their microtopography. The surface characteristics of the soil vary depending on the crops that are cultivated in these areas. Agricultural soil microtopography plays an important role in the quantification of runoff and sediment transport because the presence of crops, crop residues, furrows and ridges may impact the direction of water flow. To better assess such phenomena, 3-D reconstructions of high-resolution agricultural watershed topography are essential. Fine-resolution topographic data collection technologies can be used to discern highly detailed elevation variability in these areas. Knowledge of the strengths and weaknesses of existing technologies used for data collection on agricultural watersheds may be helpful in choosing an appropriate technology. This study assesses the suitability of terrestrial laser scanning (TLS) and unmanned aerial system (UAS) photogrammetry for collecting the fine-resolution topographic data required to generate accurate, high-resolution digital elevation models (DEMs) in a small watershed area (12ha). Because of farming activity, 14 TLS scans (≈25pointsm−2) were collected without using high-definition surveying (HDS) targets, which are generally used to mesh adjacent scans. To evaluate the accuracy of the DEMs created from the TLS scan data, 1098 ground control points (GCPs) were surveyed using a real time kinematic global positioning system (RTK-GPS). Linear regressions were then applied to each DEM to remove vertical errors from the TLS point elevations, errors caused by the non-perpendicularity of the scanner's vertical axis to the local horizontal plane, and errors correlated with the distance to the scanner's position. The scans were then meshed to generate a DEMTLS with a 1×1m spatial resolution. The Agisoft PhotoScan and MicMac software packages were used to process the aerial photographs and generate a DEMPSC (Agisoft PhotoScan) and DEMMCM (MicMac), respectively, with spatial resolutions of 1×1m. Comparing the DEMs with the 1098 GCPs showed that the DEMTLS was the most accurate data product, with a root mean square error (RMSE) of 4.5cm, followed by the DEMMCM and the DEMPSC, which had RMSE values of 9.0 and 13.9cm, respectively. The DEMPSC had absolute errors along the border of the study area that ranged from 15.0 to 52.0cm, indicating the presence of systematic errors. Although the derived DEMMCM was accurate, an error analysis along a transect showed that the errors in the DEMMCM data tended to increase in areas of lower elevation. Compared with TLS, UAS is a promising tool for data collection because of its flexibility and low operational cost. However, improvements are needed in the photogrammetric processing of the aerial photographs to remove non-linear distortions.

Measuring the Radius of the Earth from a Mountain Top Overlooking the Ocean

A clear view of the ocean may be used to measure the radius of the Earth. To an observer looking out at the ocean, the horizon will always form some angle θ with the local horizontal plane. As the observer's elevation h increases, so does the angle θ (see Fig. 1). From measurements of the elevation h and the angle θ, the radius R of the Earth may be found. This paper describes an experiment in which measurements were taken at an elevation of approximately 485 m in the Santa Monica hills of Los Angeles, CA. The average refraction-corrected value for the radius of the (assumed spherical) Earth was found to be 6100 ± 420 km. The accepted value at the equator is 6380 km.

Spacecraft formation-keeping using a closed-form orbit propagator and optimization technique

In this paper, a simple method for modeling the relative orbital motion of multiple spacecraft and their formation-keeping control strategy is presented. Power series and trigonometric functions are used to express the relative orbital motion between the member spacecraft. Their coefficients are obtained using least square regression such that the difference between the exact numerically integrated position vector and the approximate vector obtained from the closed-form propagator is minimized. Then, this closed-form orbit propagator and optimization technique is used to plan a series of impulsive maneuvers which maintain the formation configuration within a specified limit. As an example, formation-keeping of four spacecraft is investigated. The motion projected onto the local horizontal plane (along- and cross-track plane) is a circle with the leader satellite located at its center and follower satellites positioned circumferentially. The radial distance between the leader and the followers, and the relative phase angles between the followers are controlled. Results from the nonlinear simulation are presented.

楕円軌道のフォーメーションフライトに関する考察

Formation flight of spacecraft in eccentric orbits is considered in this paper. Symmetry of the eccentric orbits is assumed in a relative formation orbit, and under this assumption, relations of initial conditions in formation flight are derived. Some characteristic orbits are obtained from these relations, such as an orbit where a deputy spacecraft rotates around a chief spacecraft with relatively small variation in the distance between the two spacecraft, or an orbit where the deputy spacecraft moves on a local horizontal plane on which the chief spacecraft exists. Formation flight by plural spacecraft is possible by combining these relative orbits.

Three-axis air-bearing based platform for small satellite attitude determination and control simulation

A frictionless environment simulation platform, utilized for accomplishing three-axis attitude control tests in small satellites, is introduced. It is employed to develop, improve, and carry out objective tests of sensors, actuators, and algorithms in the experimental framework. Different sensors (i.e. sun, earth, magnetometer, and an inertial measurement unit) are utilized to assess three-axis deviations. A set of three inertial wheels is used as primary actuators for attitude control, together with three mutually perpendicular magnetic coils intended for desaturation purposes, and as a backup control system. Accurate balancing, through the platform’s center of mass relocation into the geometrical center of the spherical air-bearing, significatively reduces gravitational torques, generating a virtually torque-free environment. A very practical balancing procedure was developed for equilibrating the table in the local horizontal plane, with a reduced final residual torque. A wireless monitoring system was developed for on-line and post-processing analysis; attitude data are displayed and stored, allowing properly evaluate the sensors, actuators, and algorithms. A specifically designed onboard computer and a set of microcontrollers are used to carry out attitude determination and control tasks in a distributed control scheme. The main components and subsystems of the simulation platform are described in detail.

Counterstreaming in a Large Polar Crown Filament

The motion of small-scale structures is well resolved in high-resolution filament images that were observed on 19 June 1998 with the Swedish Vacuum Solar Telescope, La Palma. The filament was between 80 000 and 100 000 km high. The study is based on two hours of narrow-band observations at three wavelength positions in Hα. Velocities along the line of sight and in the transverse direction, respectively, V los and V tr, were measured for a large number of individual small-scale filament structures. Small features are all moving along nearly parallel threads, some in one direction along the threads and the remainder in the other direction, a pattern of motion known as counterstreaming. The net flow velocities in the two directions are about 8 km s−1 and both are tilted by an angle δ≃16° relative to the plane of the sky. This angle is less than expected, by factors between 2.0 and 2.5, relative to the local horizontal plane. We believe that V los is underestimated by these factors due to a line-shift reducing effect by the underlying Hα absorption line of the chromosphere.

Guidance and Relative Navigation for Autonomous Rendezvous in a Circular Orbit

Algorithmsforautonomousguidanceofspacecraftto approach,toe yaround,andtodepartfroma targetvehicle ina circularorbitarepresented. Thealgorithmsarebasedon theclosed-form solution of linearClohessy ‐Wiltshire equations. The approach and departure algorithms are adaptations of the glideslope guidance used in the past for rendezvous and proximity operations of the space shuttle with other vehicles with astronauts in the guidance loop. The multipulse glideslope algorithms are general, capable of effecting a translation motion of spacecraft in any direction in space autonomously, decelerating while approaching a target or a nearby location, and accelerating while receding. The e yaround algorithm enables the spacecraft to circumnavigate a target spacecraft in any plane, the orbit plane and the local horizontal plane being two special cases thereof. The circumnavigation is performed in a specie ed period using a specie ed number of pulses; the larger the number of pulses, the smaller the deviation of e yaround from the specie ed radius of circumnavigation. The implementation of these algorithms requires estimates of position and velocity of the spacecraft relative to the target. This relative navigation is performed with an extended Kalman e lter using range and angle measurements of the target relative to the spacecraft focal plane and the spacecraft attitude estimates from an inertial navigation system. The corresponding measurement models and process noise matrix are provided. Several scenarios are simulated to illustrate the guidance algorithms and relative navigation.

Tidally induced Sagnac signal in a ring laser

The Earth rotation vector is an integral dynamical quantity of the Earth system, with Earth rotation variations depending on the overall dynamics of the Earth system at all time scales. New observations of Earth rotation variations with a precision better than 10−9 of the Earth rotation would contribute to the understanding of many geophysical phenomena. High precision short‐term observations of the projection of the Earth rotation vector ΩE on the area vector A of a ring laser are now possible. Tidally induced Sagnac effects are due to changes in the ring geometry (areal strain and perimeter variations), changes in the normal vector of the ring laser plane (tilt), and the vorticity of the Earth's tidal deformations. Here we estimate the latitude‐dependent effect of the Earth's body tide deformations due to vorticity and tilt. At mid‐latitudes, the vorticity δΩ is of the order 8 · 10−8ΩE, but because it is in the local horizontal plane, it is not detectable by a horizontal ring. The Sagnac effect due to tidal tilt for a horizontal ring at mid‐latitudes is of the order of 4 · 10−8 of the Earth rotation effect. Thus, the tidal signal is of an order of magnitude where it may be used for validation of existing and future ring lasers.

VLBI data analysis with the GEOSAT software

The GEOSAT software for satellite geodetic and geodynamics applications has lately been extended to Very Long Baseline Interferometry (VLBI) applications. Geodetic VLBI data have been analyzed using this software and station coordinates and velocities, source coordinates, and earth orientation parameters have been derived in a global mode using data from the Extended Research and Development Experiment (ERDE) and the Research and Development series (R&D) within the NASA's CDP. It is demonstrated using the ERDE data, that the program is capable of calculating station coordinates with an accuracy of a few mm in the local horizontal plane and about 7 mm in the vertical direction. Analysis of the R&D dataset yields errors 2 to 3 times higher in all directions.