Colatitude

Abstract: The conventional expansions of the gravity gradients in the local north-oriented reference frame have a complicated form, depending on the first- and second-order derivatives of the associated Legendre functions of the colatitude and containing factors which tend to infinity when approaching the poles. In the present paper, the general term of each of these series is transformed to a product of a geopotential coefficient $$\overline{C}_{n,m} $$ and a sum of several adjacent Legendre functions of the colatitude multiplied by a function of the longitude. These transformations are performed on the basis of relations between the Legendre functions and their derivatives published by Ilk (1983). The second-order geopotential derivatives corresponding to the local orbital reference frame are presented as linear functions of the north-oriented gravity gradients. The new expansions for the latter are substituted into these functions. As a result, the orbital derivatives are also presented as series depending on the geopotential coefficients $$\overline{C}_{n,m} $$ multiplied by sums of the Legendre functions whose coefficients depend on the longitude and the satellite track azimuth at an observation point. The derived expansions of the observables can be applied for constructing a geopotential model from the GOCE mission data by the time-wise and space-wise approaches. The numerical experiments demonstrate the correctness of the analytical formulas.

Abstract: A new technique, the method of spherical cap harmonic analysis, has provided the means to construct an analytical model of the vertical field above 40°N from Magsat data. With this technique, proper consideration can be taken of altitude variation and of the source-free constraint that the vertical field be the vertical derivative of a potential function. Maps can be produced at any altitude. Here they are given at the lower and upper limits of the Magsat satellite, 6700 and 6900 km radial distance or approximately 335 and 535 km geodetic altitude. A novel decimation procedure, applied to NASA “Investigator B” data screened for low disturbance levels, results in an approximately uniform distribution in area, about 1 point per (150 km)2. Each of the 5087 data points represents an average over 5 s of time or 37 km of satellite track. The harmonic model, of maximum index 22, includes only those basis functions that are necessary for fitting a potential which need not be differentiable in colatitude and that are therefore completely orthogonal over the spherical cap. Although there are 276 coefficients in the theoretical model, only 190 were found to be statistically significant. The resulting scatter, or standard error of estimate, was 3.6 nT.

Abstract: Modified Fourier series, as judged by criteria of accuracy, numerical efficiency and ease of programming, are the best choice of latitudinal expansion functions for general problems on the sphere. The pseudospectral and spectral methods, however, can be easily and successfully applied with all three types of orthogonal series. For special situations, such as when the latitude variable is stretched, Chebyshev polynomials are the only practical choice, but for orthodox problems on the globe, they are less efficient than the other two sets of functions. Although spherical harmonics have been universally employed in the past, Fourier series give comparable accuracy and are significantly easier to program and manipulate. Thus, in the absence of a special reason to the contrary, the simplest and most effective way to handle the north–south dependence of the solution to a boundary or eigenvalue problem on the sphere is to use a Fourier series in colatitude.

Abstract: This paper presents a formulation of image-based visual servoing (IBVS) for a spherical camera where coordinates are parameterized in terms of colatitude and longitude: IBVS-Sph. The image Jacobian is derived and simulation results are presented for canonical rotational, translational as well as general motion. Problems with large rotations that affect the planar perspective form of IBVS are not present on the sphere, whereas the desirable robustness properties of IBVS are shown to be retained. We also describe a structure from motion (SfM) system based on camera-centric spherical coordinates and show how a recursive estimator can be used to recover structure. The spherical formulations for IBVS and SfM are particularly suitable for platforms, such as aerial and underwater robots, that move in SE(3).