Planning algorithms are impacting technical disciplines and industries around the world, including robotics, computer-aided design, manufacturing, computer graphics, aerospace applications, drug design, and protein folding. This coherent and comprehensive book unifies material from several sources, including robotics, control theory, artificial intelligence, and algorithms. The treatment is centered on robot motion planning but integrates material on planning in discrete spaces. A major part of the book is devoted to planning under uncertainty, including decision theory, Markov decision processes, and information spaces, which are the “configuration spaces” of all sensor-based planning problems. The last part of the book delves into planning under differential constraints that arise when automating the motions of virtually any mechanical system. Developed from courses taught by the author, the book is intended for students, engineers, and researchers in robotics, artificial intelligence, and control theory as well as computer graphics, algorithms, and computational biology.

Planning Algorithms: Introductory Material

A data set comprising 110 spreading rates, 78 transform fault azimuths, and 142 earthquake slip vectors has been inverted to yield a new instantaneous plate motion model, designated Relative Motion 2 (RM2). The model represents a considerable improvement over our previous estimate, RM1 [Minster et al., 1974]. The mean averaging interval for the spreading rate data has been reduced to less than 3 m.y. A detailed comparison of RM2 with angular velocity vectors which best fit the data along individual plate boundaries indicates that RM2 performs close to optimally in most regions, with several notable exceptions. The model systematically misfits data along the India-Antarctica and Pacific-India plate boundaries. We hypothesize that these discrepancies are manifestations of internal deformation within the Indian plate; the data are compatible with northwest-southeast compression across the Ninetyeast Ridge at a rate of about 1 cm/yr. RM2 also fails to satisfy the east-west trending transform fault azimuths observed in the French-American Mid-Ocean Undersea Study area, which is shown to be a consequence of closure constraints about the Azores triple junction. Slow movement between North and South America is required by the data set, although the angular velocity vector describing this motion remains poorly constrained. The existence of a Bering plate, postulated in our previous study, is not necessary if we accept the proposal of Engdahl and others that the Aleutian slip vector data are biased by slab effects. Absolute motion models are derived from several kinematical hypotheses and compared with the data from hot spot traces younger than 10 m.y. Although some of the models are inconsistent with the Wilson-Morgan hypothesis, the overall resolving power of the hot spot data is poor, and the directions of absolute motion for the several slower-moving plates are not usefully constrained.

Present-day plate motions

A data set comprising 110 spreading rates, 78 transform fault azimuths and 142 earthquake slip vectors was inverted to yield a new instantaneous plate motion model, designated RM2. The mean averaging interval for the relative motion data was reduced to less than 3 My. A detailed comparison of RM2 with angular velocity vectors which best fit the data along individual plate boundaries indicates that RM2 performs close to optimally in most regions, with several notable exceptions. On the other hand, a previous estimate (RM1) failed to satisfy an extensive set of new data collected in the South Atlantic Ocean. It is shown that RM1 incorrectly predicts the plate kinematics in the South Atlantic because the presently available data are inconsistent with the plate geometry assumed in deriving RM1. It is demonstrated that this inconsistency can be remedied by postulating the existence of internal deformation with the Indian plate, although alternate explanations are possible.

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Present‐day plate motions

The composition of chromian spinel in alpine-type peridotites has a large reciprocal range of Cr and Al, with increasing Cr# (Cr/(Cr+Al)) reflecting increasing degrees of partial melting in the mantle. Using spinel compositions, alpine-type peridotites can be divided into three groups. Type I peridotites and associated volcanic rocks contain spinels with Cr# 0.60, and Type II peridotites and volcanics are a transitional group and contain spinels spanning the full range of spinel compositions in Type I and Type II peridotites. Spinels in abyssal peridotites lie entirely within the Type I spinel field, making ophiolites with Type I alpine-type peridotites the most likely candidates for sections of ocean lithosphere formed at a midocean ridge. The only modern analogs for Type III peridotites and associated volcanic rocks are found in arc-related volcanic and intrusive rocks, continental intrusive assemblages, and oceanic plateau basalts. We infer a sub-volcanic arc petrogenesis for most Type III alpine-type peridotites. Type II alpine-type peridotites apparently reflect composite origins, such as the formation of an island-arc on ocean crust, resulting in large variations in the degree and provenance of melting over relatively short distances. The essential difference between Type I and Type III peridotites appears to be the presence or absence of diopside in the residue at the end of melting.

Chromian spinel as a petrogenetic indicator in abyssal and alpine-type peridotites and spatially associated lavas

Regional averages of the major element chemistry of ocean ridge basalts, corrected for low-pressure fractionation, correlate with regional averages of axial depth for the global system of ocean ridges, including hot spots, cold spots, and back arc basins, as well as “normal” ocean ridges. Quantitative consideration of the variations of each major element during melting of the mantle suggests that the global major element variations can be accounted for by ∼8–20% melting of the mantle at associated mean pressures of 5–16 kbar. The lowest extents of melting occur at shallowest depths in the mantle and are associated with the deepest ocean ridges. Calculated mean primary magmas show a range in composition from 10 to 15 wt % MgO, and the primary magma compositions correlate with depth. Data for Sm, Yb, Sc, and Ni are consistent with the major elements, but highly incompatible elements show more complicated behavior. In addition, some hot spots have anomalous chemistry, suggesting major element heterogeneity. Thermal modeling of mantle ascending adiabatically beneath the ridge is consistent with the chemical data and melting calculations, provided the melt is tapped from throughout the ascending mantle column. The thermal modeling independently predicts the observed relationships among basalt chemistry, ridge depth, and crustal thickness resulting from temperature variations in the mantle. Beneath the shallowest and deepest ridge axes, temperature differences of approximately 250°C in the subsolidus mantle are required to account for the global systematics.

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Global correlations of ocean ridge basalt chemistry with axial depth and crustal thickness

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/pdf/the-foundations-of-mechanics-24bf2o7xcj.pdf

The foundations of mechanics.

A simplified model is proposed that can explain the evolution of marginal offsets, ridges, and fracture zones. Submarine features of the equatorial Atlantic margins are examined and are found to be in reasonable agreement with the model. The results reaffirm the suggestion that the South Atlantic opened in two stages beginning about 140 m.y. and 80 m.y. ago. The pole of rotation describing the relative motion of the South American and African plates changed radically when the constraints imposed by adjacent continental blocks were relaxed.

Marginal offsets, fracture zones, and the early opening of the North Atlantic

Massive ore-grade zinc, copper and iron sulphide deposits have been found at the axis of the East Pacific Rise. Although their presence on the deep ocean-floor had been predicted there was no supporting observational evidence. The East Pacific Rise deposits represent a modern analogue of Cyprus-type sulphide ores associated with ophiolitic rocks on land. They contain at least 29% zinc metal and 6% metallic copper. Their discovery will provide a new focus for deep-sea exploration, leading to new assessments of the concentration of metals in the upper layers of the oceanic crust.

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Massive deep-sea sulphide ore deposits discovered on the East Pacific Rise

A simple model of the probable topographic and thermal consequences of subducting an oceanic spreading center at an island arc predicts three geologic effects: shoaling and subaerial emergence of the crest of the arc; decrease or cessation of subduction-related magmatism; and regional low-grade thermal metamorphism (..delta..T approximately equal to 100 to 300/sup 0/C) of the arc rocks. All three of these phenomena are recorded in the geology of the Aleutian Islands, and the following sequence of events is indicated: (1) diminution of magmatism on approach of the Kula Ridge in middle Eocene times (approximately equal to 45 m.y. ago), marked by a conformable transition from volcanic-rich to volcanic-poor early-series rocks; (2) shoaling and emergence of the crest of the Aleutian arc in late Eocene to Oligocene time, marked by a deep- to shallow-marine transition in sedimentation and then an arc-wide unconformity above the early series and its probable ''submarine equivalent,'' the now-dissected Aleutian crestal platform; (3) subduction of the Kula Ridge and greenschist metamorphism of the early-series rocks at about 30 to 35 m.y. ago, inferred from K-Ar ages; (4) subsidence of the arc down the south flank of the Kula Ridge in middle Oligocene to Miocene time, as themore » Pacific plate was subducted; and (5) abrupt resumption of arc magmatism 15 m.y. ago. This history of events is consistent with the timing of plate motions in the North Pacific and suggests that there has been essentially continuous underthrusting at the Aleutian Trench since at least 45 to 50 m.y. ago, with subduction of 900 to 1,000 km of ocean floor since 30 to 35 m.y. ago.« less

Subduction of the Kula Ridge at the Aleutian Trench

THE significance and origin of ophiolites is a matter of interest to geologists. Many investigators interpret ophiolites as slices of oceanic crust which have been technically emplaced in orogenic belts1–7. Moores and Vine5, Dewey and Bird6, and Coleman7 have proposed just such an origin for ophiolites as geographically diverse as Cyprus, Newfoundland and the Western United States. If ophiolites are representative of the geology of the oceanic crust, then they provide direct access to samples of oceanic crust and the underlying upper mantle—rocks at present inaccessible except for the fortuitous catch in dredge hauls located on tectonic escarpments of the ocean floor. Although we accepted the proposition that ophiolites, if well preserved and not altered by events unrelated to lithosphere accretion, are representative of oceanic crust, we wished first to test this proposition by observing whether representative ophiolite samples had measured velocities compatible with the seismic layering defined for oceanic crust. If the velocities of ophiolite samples were compatible with the range of seismic velocities recorded for oceanic crust, then we hoped to use the ophiolite measurements to understand better the details of the seismic layering of oceanic crust. Here we summarise the acoustic velocity data obtained from metadolerites, gabbros and cumulate gabbros from the Mings Bight ophiolite assemblage6 located along the northern coast of Newfoundland. A more detailed discussion of these ophiolite samples (J. J. P., in preparation) will give a thorough presentation of petrography, sample density and measured compressional wave velocity of each sample at pressure intervals from 1 atm to 7 kbar.

Paul J. Fox

Papers

The foundations of mechanics.

Marginal offsets, fracture zones, and the early opening of the North Atlantic

Massive deep-sea sulphide ore deposits discovered on the East Pacific Rise

Subduction of the Kula Ridge at the Aleutian Trench

Newfoundland Ophiolites and the Geology of the Oceanic Layer