Optimal steady motions for oriented vehicles

Motivated by the motion planning problem for oriented vehicles travelling in a 3-Dimensional space; Euclidean space E3, the sphere S3 and Hyperboloid H3. For such problems the orientation of the vehicle is naturally represented by an orthonormal frame over a point in the underlying manifold. The orthonormal frame bundles of the space forms R3,S3 and H3 correspond with their isometry groups and are the Euclidean group of motion SE(3), the rotation group SO(4) and the Lorentzian group SO(1; 3) respectively. Orthonormal frame bundles of space forms coincide with their isometry groups and therefore the focus shifts to left-invariant control systems defined on Lie groups. In this paper a method for integrating these systems is given where the controls are time-independent. For constant twist motions or helical motions, the corresponding curves g(t) 2 SE(3) are given in closed form by using the well known Rodrigues’ formula. However, this formula is only applicable to the Euclidean case. This paper gives a method for computing the non-Euclidean screw/helical motions in closed form. This involves decoupling the system into two lower dimensional systems using the double cover properties of Lie groups, then the lower dimensional systems are solved explicitly in closed form.

Walking in a winter wonderland? Strategies for Early and Middle Pleistocene survival in mid-latitude Europe

Any occupation of northern Europe by Lower Palaeolithic hominins, even those occurring during full interglacials, must have addressed the challenges of marked seasonality and cold winters. These would have included the problems of: wind-chill and frostbite; duration, distribution and depth of snow-cover; reduced daylight hours; and distribution and availability of animal and plant foods. Solutions can essentially be characterised as a ‘stick or twist’ choice: i.e. year-round presence on a local scale vs. extensive annual mobility. However these options, and the ‘interim’ strategies that lie between them, present various problems, including maintaining core body temperature, meeting the energetic demands of mobility, coping with reduced resource availability and increasing patchiness, and meeting nutritional requirements. The feasibility of different winter survival strategies are explored with reference to Lower Palaeolithic palaeoenvironmental reconstructions and on-site behavioural evidence. Emphasis is placed upon possible strategies for (i) avoiding the excessive lean meat protein problem of ‘rabbit starvation’ (e.g. through exploitation of ‘residential’ species with significant winter body fat and/or by targeting specific body parts, following modern ethnographic examples, supplemented by the exploitation of winter plants); and (ii) maintaining body temperatures (e.g. through managed pyrotechnology, and/or other forms of cultural insulation). The paper concludes with a suggested winter strategy.