Singularities of optimal control problems on some 6-D lie groups

This paper considers the motion planning problem for oriented vehicles travelling at unit speed in a 3-D space. A Lie group formulation arises naturally and the vehicles are modeled as kinematic control systems with drift defined on the orthonormal frame bundles of particular Riemannian manifolds, specifically, the 3-D space forms Euclidean space E-3, the sphere S-3, and the hyperboloid H'. The corresponding frame bundles are equal to the Euclidean group of motions SE(3), the rotation group SO(4), and the Lorentz group SO (1, 3). The maximum principle of optimal control shifts the emphasis for these systems to the associated Hamiltonian formalism. For an integrable case, the extremal curves are explicitly expressed in terms of elliptic functions. In this paper, a study at the singularities of the extremal curves are given, which correspond to critical points of these elliptic functions. The extremal curves are characterized as the intersections of invariant surfaces and are illustrated graphically at the singular points. It. is then shown that the projections, of the extremals onto the base space, called elastica, at these singular points, are curves of constant curvature and torsion, which in turn implies that the oriented vehicles trace helices.

Energy at work: a measurement validation and linkage to unit effectiveness

We introduce the notion that the energy of individuals can manifest as a higher-level, collective construct. To this end, we conducted four independent studies to investigate the viability and importance of the collective energy construct as assessed by a new survey instrument—the productive energy measure (PEM). Study 1 (n = 2208) included exploratory and confirmatory factor analyses to explore the underlying factor structure of PEM. Study 2 (n = 660) cross-validated the same factor structure in an independent sample. In study 3, we administered the PEM to more than 5000 employees from 145 departments located in five countries. Results from measurement invariance, statistical aggregation, convergent, and discriminant-validity assessments offered additional support for the construct validity of PEM. In terms of predictive and incremental validity, the PEM was positively associated with three collective attitudes—units' commitment to goals, the organization, and overall satisfaction. In study 4, we explored the relationship between the productive energy of firms and their overall performance. Using data from 92 firms (n = 5939employees), we found a positive relationship between the PEM (aggregated to the firm level) and the performance of those firms. Copyright © 2011 John Wiley & Sons, Ltd.