Planning rigid body motions using elastic curves

This paper tackles the problem of computing smooth, optimal trajectories on the Euclidean group of motions SE(3). The problem is formulated as an optimal control problem where the cost function to be minimized is equal to the integral of the classical curvature squared. This problem is analogous to the elastic problem from differential geometry and thus the resulting rigid body motions will trace elastic curves. An application of the Maximum Principle to this optimal control problem shifts the emphasis to the language of symplectic geometry and to the associated Hamiltonian formalism. This results in a system of first order differential equations that yield coordinate free necessary conditions for optimality for these curves. From these necessary conditions we identify an integrable case and these particular set of curves are solved analytically. These analytic solutions provide interpolating curves between an initial given position and orientation and a desired position and orientation that would be useful in motion planning for systems such as robotic manipulators and autonomous-oriented vehicles.

Finding the point on Bezier Curves with the normal vector passing an external point

Two algorithms for finding the point on non-rational/rational Bezier curves of which the normal vector passes through a given external point are presented. The algorithms are based on Bezier curves generation algorithms of de Casteljau's algorithm for non-rational Bezier curve or Farin's recursion for rational Bezier curve, respectively. Orthogonal projections from the external point are used to guide the directional search used in the proposed iterative algorithms. Using Lyapunov's method, it is shown that each algorithm is able to converge to a local minimum for each case of non-rational/rational Bezier curves. It is also shown that on convergence the distance between the point on curves to the external point reaches a local minimum for both approaches. Illustrative examples are included to demonstrate the effectiveness of the proposed approaches.

Use of Megasphaera elsdenii NCIMB41125 as a probiotic for early-lactation dairy cows: effects on rumen pH and fermentation patterns

Multiparous rumen-fistulated Holstein cows were fed, from d 1 to 28 post-calving, an ad libitum TMR containing (g/kg DM) grass silage (196), corn silage (196), wheat (277), soybean meal (100), and other feeds (231) with CP, NDF, starch and water soluble carbohydrate concentrations of 176, 260, 299 and 39 g/kg DM respectively and ME of 12.2 MJ/kg DM. Treatments consisting of a minimum of 1010 cfu Megasphaera elsdenii NCIMB 41125 in 250 ml solution (MEGA) or 250 ml of autoclaved M. elsdenii (CONT) were administered via the rumen cannula on d 3 and 12 of lactation (n=7 per treatment). Mid-rumen pH was measured every 15 minutes and eating and ruminating behavior was recorded for 24 h on d 2, 4, 6, 8, 11, 13, 15, 17, 22 and 28. Rumen fluid for VFA and lactic acid (LA) analysis was collected at 11 timepoints on each of d 2, 4, 6, 13 and 15. Data were analysed as repeated measures using the Glimmix (LA data) or Mixed (all other data) procedures of SAS with previous 305 d milk yield and d 2 measurements as covariates where appropriate. Milk yield was higher (CONT 43.0 vs MEGA 45.4 ±0.75 kg/d, P=0.051) and fat concentration was lower (CONT 45.6 vs MEGA 40.4 ±1.05 g/kg, P=0.005) in cows that received MEGA. Time spent eating (263 ±15 min/d) and ruminating (571 ±13 min/d), DM intake (18.4 ±0.74 kg/d), proportion of each 24 h period with rumen pH below 5.6 (3.69 ±0.94 h) and LA concentrations (2.00 mM) were similar (P>0.327) across treatments. Ruminal total VFA concentration (104 ±3 mM) was similar (P=0.404) across treatments, but a shift from acetate (CONT 551 vs MEGA 524 ±14 mmol/mol VFA, P=0.161) to propionate production (CONT 249 vs MEGA 275 ±11 mmol/mol VFA, P=0.099) meant that the acetate:propionate ratio (CONT 2.33 vs MEGA 1.94 ±0.15) was reduced (P=0.072) in cows that received MEGA. This study provides evidence that supplementation of early lactation dairy cows with MEGA alters rumen fermentation patterns in favour of propionate, with potential benefits for animal health and productivity.

The elasticity of demand is about occupancy, not prices: building European demand curves based on occupancy elasticity

The peak congestion of the European grid may create significant impacts on system costs because of the need for higher marginal cost generation, higher cost system balancing and increasing grid reinforcement investment. The use of time of use rates, incentives, real time pricing and other programmes, usually defined as Demand Side Management (DSM), could bring about significant reductions in prices, limit carbon emissions from dirty power plants, and improve the integration of renewable sources of energy. Unlike previous studies on elasticity of residential electricity demand under flat tariffs, the aim of this study is not to investigate the known relatively inelastic relationship between demand and prices. Rather, the aim is to assess how occupancy levels vary in different European countries. This reflects the reality of demand loads, which are predominantly determined by the timing of human activities (e.g. travelling to work, taking children to school) rather than prices. To this end, two types of occupancy elasticity are estimated: baseline occupancy elasticity and peak occupancy elasticity. These represent the intrinsic elasticity associated with human activities of single residential end-users in 15 European countries. This study makes use of occupancy time-series data from the Harmonised European Time Use Survey database to build European occupancy curves; identify peak occupancy periods; draw time use demand curves for video and TV watching activity; and estimate national occupancy elasticity levels of single-occupant households. Findings on occupancy elasticities provide an indication of possible DSM strategies based on occupancy levels and not prices.

Anodic polarization curves revisited

An experiment published in this Journal has been revisited and it is found that the curve pattern of the anodic polarization curve for iron repeats itself successively when the potential scan is repeated. It is surprising that this observation has not been reported previously in the literature because it immediately brings into question the long accepted and well-known explanations involving a passive film. A qualitative and plausible explanation is provided from surprisingly simple principles for this new finding. Some important pedagogic conclusions have been derived from this work. It is noteworthy that the somewhat complicated phenomenon can be simply explained, thus providing two important lessons to students. First, even well-accepted scientific work studying simple processes may be incomplete and worthy of further study, and second, such processes may be explained simply at the undergraduate level. The contents of the paper also confirm that presenting curricular contents in a new and more correct manner is beneficial, interesting, and that research in curricular contents represents one of the important forms of educational research.