Optimal kinematic control of an autonomous underwater vehicle

This note investigates the motion control of an autonomous underwater vehicle (AUV). The AUV is modeled as a nonholonomic system as any lateral motion of a conventional, slender AUV is quickly damped out. The problem is formulated as an optimal kinematic control problem on the Euclidean Group of Motions SE(3), where the cost function to be minimized is equal to the integral of a quadratic function of the velocity components. An application of the Maximum Principle to this optimal control problem yields the appropriate Hamiltonian and the corresponding vector fields give the necessary conditions for optimality. For a special case of the cost function, the necessary conditions for optimality can be characterized more easily and we proceed to investigate its solutions. Finally, it is shown that a particular set of optimal motions trace helical paths. Throughout this note we highlight a particular case where the quadratic cost function is weighted in such a way that it equates to the Lagrangian (kinetic energy) of the AUV. For this case, the regular extremal curves are constrained to equate to the AUV's components of momentum and the resulting vector fields are the d'Alembert-Lagrange equations in Hamiltonian form.

Palaeohydrology of the Fazzan Basin and surrounding regions: The last 7 million years

We have integrated information on topography, geology and geomorphology with the results of targeted fieldwork in order to develop a chronology for the development of Lake Megafazzan, a giant lake that has periodically existed in the Fazzan Basin since the late Miocene. The development of the basin can be best understood by considering the main geological and geomorphological events that occurred thought Libya during this period and thus an overview of the palaeohydrology of all Libya is also presented. The origin of the Fazzan Basin appears to lie in the Late Miocene. At this time Libya was dominated by two large rivers systems that flowed into the Mediterranean Sea, the Sahabi River draining central and eastern Libya and the Wadi Nashu River draining much of western Libya. As the Miocene progressed the region become increasingly affected by volcanic activity on its northern and eastern margin that appears to have blocked the River Nashu in Late Miocene or early Messinian times forming a sizeable closed basin in the Fazzan within which proto-Lake Megafazzan would have developed during humid periods. The fall in base level associated with the Messinian desiccation of the Mediterranean Sea promoted down-cutting and extension of river systems throughout much of Libya. To the south of the proto Fazzan Basin the Sahabi River tributary know as Wadi Barjuj appears to have expanded its headwaters westwards. The channel now terminates at Al Haruj al Aswad. We interpret this as a suggestion that Wadi Barjuj was blocked by the progressive development of Al Haruj al Aswad. K/Ar dating of lava flows suggests that this occurred between 4 and 2 Ma. This event would have increased the size of the closed basin in the Fazzan by about half, producing a catchment close to its current size (-350,000 km(2)). The Fazzan Basin contains a wealth of Pleistocene to recent palaeolake sediment outcrops and shorelines. Dating of these features demonstrates evidence of lacustrine conditions during numerous interglacials spanning a period greater than 420 ka. The middle to late Pleistocene interglacials were humid enough to produce a giant lake of about 135,000 km(2) that we have called Lake Megafazzan. Later lake phases were smaller, the interglacials less humid, developing lakes of a few thousand square kilometres. In parallel with these palaeohydrological developments in the Fazzan Basin, change was occurring in other parts of Libya. The Lower Pliocene sea level rise caused sediments to infill much of the Messinian channel system. As this was occurring, subsidence in the Al Kufrah Basin caused expansion of the Al Kufrah River system at the expense of the River Sahabi. By the Pleistocene, the Al Kufrah River dominated the palaeohydrology of eastern Libya and had developed a very large inland delta in its northern reaches that exhibited a complex distributary channel network which at times fed substantial lakes in the Sirt Basin. At this time Libya was a veritable lake district during humid periods with about 10% of the country underwater. (C) 2008 Elsevier B.V. All rights reserved.

Computer modelling and estimation of recruitment patterns of non-branching coral colonies at three sites in Wakatobi Marine Park, S.E. Sulawesi, Indonesia; implications for coral reef conservation

We have studied growth and estimated recruitment of massive coral colonies at three sites, Kaledupa, Hoga and Sampela, separated by about 1.5 km in the Wakatobi Marine National Park, S.E. Sulawesi, Indonesia. There was significantly higher species richness (P<0.05), coral cover (P<0.05) and rugosity (P<0.01) at Kaledupa than at Sampela. A model for coral reef growth has been developed based on a rational polynomial function, where dx/dt is an index of coral growth with time; W is the variable (for example, coral weight, coral length or coral area), up to the power of n in the numerator and m in the denominator; a1……an and b1…bm are constants. The values for n and m represent the degree of the polynomial, and can relate to the morphology of the coral. The model was used to simulate typical coral growth curves, and tested using published data obtained by weighing coral colonies underwater in reefs on the south-west coast of Curaçao [‘Neth. J. Sea Res. 10 (1976) 285’]. The model proved an accurate fit to the data, and parameters were obtained for a number of coral species. Surface area data was obtained on over 1200 massive corals at three different sites in the Wakatobi Marine National Park, S.E. Sulawesi, Indonesia. The year of an individual's recruitment was calculated from knowledge of the growth rate modified by application of the rational polynomial model. The estimated pattern of recruitment was variable, with little numbers of massive corals settling and growing before 1950 at the heavily used site, Sampela, relative to the reef site with little or no human use, Kaledupa, and the intermediate site, Hoga. There was a significantly greater sedimentation rate at Sampela than at either Kaledupa (P<0.0001) or Hoga (P<0.0005). The relative mean abundance of fish families present at the reef crests at the three sites, determined using digital video photography, did not correlate with sedimentation rates, underwater visibility or lack of large non-branching coral colonies. Radial growth rates of three genera of non-branching corals were significantly lower at Sampela than at Kaledupa or at Hoga, and there was a high correlation (r=0.89) between radial growth rates and underwater visibility. Porites spp. was the most abundant coral over all the sites and at all depths followed by Favites (P<0.04) and Favia spp. (P<0.03). Colony ages of Porites corals were significantly lower at the 5 m reef flat on the Sampela reef than at the same depth on both other reefs (P<0.005). At Sampela, only 2.8% of corals on the 5 m reef crest are of a size to have survived from before 1950. The Scleractinian coral community of Sampela is severely impacted by depositing sediments which can lead to the suffocation of corals, whilst also decreasing light penetration resulting in decreased growth and calcification rates. The net loss of material from Sampela, if not checked, could result in the loss of this protective barrier which would be to the detriment of the sublittoral sand flats and hence the Sampela village.

Integrable quadratic Hamiltonians on the Euclidean group of motions

In this paper, we discuss the problem of globally computing sub-Riemannian curves on the Euclidean group of motions SE(3). In particular, we derive a global result for special sub-Riemannian curves whose Hamiltonian satisfies a particular condition. In this paper, sub-Riemannian curves are defined in the context of a constrained optimal control problem. The maximum principle is then applied to this problem to yield an appropriate left-invariant quadratic Hamiltonian. A number of integrable quadratic Hamiltonians are identified. We then proceed to derive convenient expressions for sub-Riemannian curves in SE(3) that correspond to particular extremal curves. These equations are then used to compute sub-Riemannian curves that could potentially be used for motion planning of underwater vehicles.

Modeling ocean primary production: sensitivity to spectral resolution of attenuation and absorption of light

Modeling the vertical penetration of photosynthetically active radiation (PAR) through the ocean, and its utilization by phytoplankton, is fundamental to simulating marine primary production. The variation of attenuation and absorption of light with wavelength suggests that photosynthesis should be modeled at high spectral resolution, but this is computationally expensive. To model primary production in global 3d models, a balance between computer time and accuracy is necessary. We investigate the effects of varying the spectral resolution of the underwater light field and the photosynthetic efficiency of phytoplankton (α∗), on primary production using a 1d coupled ecosystem ocean turbulence model. The model is applied at three sites in the Atlantic Ocean (CIS (∼60°N), PAP (∼50°N) and ESTOC (∼30°N)) to include the effect of different meteorological forcing and parameter sets. We also investigate three different methods for modeling α∗ – as a fixed constant, varying with both wavelength and chlorophyll concentration [Bricaud, A., Morel, A., Babin, M., Allali, K., Claustre, H., 1998. Variations of light absorption by suspended particles with chlorophyll a concentration in oceanic (case 1) waters. Analysis and implications for bio-optical models. J. Geophys. Res. 103, 31033–31044], and using a non-spectral parameterization [Anderson, T.R., 1993. A spectrally averaged model of light penetration and photosynthesis. Limnol. Oceanogr. 38, 1403–1419]. After selecting the appropriate ecosystem parameters for each of the three sites we vary the spectral resolution of light and α∗ from 1 to 61 wavebands and study the results in conjunction with the three different α∗ estimation methods. The results show modeled estimates of ocean primary productivity are highly sensitive to the degree of spectral resolution and α∗. For accurate simulations of primary production and chlorophyll distribution we recommend a spectral resolution of at least six wavebands if α∗ is a function of wavelength and chlorophyll, and three wavebands if α∗ is a fixed value.