The Blaschke-Petkantschin formula is a geometric measure decomposition of the q-fold product of Lebesgue measure on R n . Here we discuss another decomposition called polar decomposition by considering R n × · · · × R n as M n×k and using its polar decomposition. This is a generalisation of the Blaschke-Petkantschin formula and may be useful when one needs to integrate a function g :As an application we compute the moments of a Gaussian determinant.2010 Mathematics subject classification: primary 28A75; secondary 49Q15, 60D05.

No abstract

This paper proposes distributed optimal attitude consensus control for single-integrator multi rigid bodies with undirected network evolving on Special Orthogonal Group SO(3) while simultaneously guarantees the connectivity preservation property for agents using descent gradient algorithm. Since by Use of the Euclidean distance on Lie group as a measure of the energy of the state does not define and preserve the topology of SO(3); besides, solving the Hamilton-Jacobi-Bellman equation in optimal control problems shows difficulty implementing Euclidean distances and limits the results for SO(3) configuration state spaces. As a result, in this paper, the generic distance on SO(3) associated to the natural Riemannian metric structure is used. Using this structure, Firstly, a distributed potential function based consensus control law is applied to the system exploiting Riemannian distance on SO(3). Then, for relaxing some restrictive conditions, finite-time convergence, and increasing the speed of convergence the kinematic optimal control on SO(3) is considered. Referring to the proposed potential function designed in the previous section, an inverse optimal attitude consensus control problem is considered, which is solved by an inverse optimal control method. Finally, the designed method validates via two simulation examples.

A pair of bodies rolling on each other is an interesting example of nonholonomic systems in control theory. Here the controllability of rolling bodies is investigated with a global approach. By using simple geometric facts, this problem has been completely solved in the special case where one of them is a plane or a sphere.