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.
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.
In CDMA systems, the received user powers vary due to moving distance of users. Thus, the CDMA receivers consist of two stages. The first stage is the power estimator and the second one is a Multi-User Detector (MUD). Conventional methods for estimating the user powers are suitable for underor fully-loaded cases (when the number of users is less than or equal to the spreading gain). These methods fail to work for overloaded CDMA systems because of high interference among the users. Since the bandwidth is becoming more and more valuable, it is worth considering overloaded CDMA systems. In this paper, an optimum user power estimation for over-loaded CDMA systems with Gaussian inputs is proposed. We also introduce a suboptimum method with lower complexity whose performance is very close to the optimum one. We shall show that the proposed methods work for highly over-loaded systems (up to m (m + 1) /2 users for a system with only m chips). The performance of the proposed methods is demonstrated by simulations. In addition, a class of signature sets is proposed that seems to be optimum from a power estimation point of view. Additionally, an iterative estimation for binary input CDMA systems is proposed which works more accurately than the optimal Gaussian input method.
Abstract. A pair of bodies rolling on each other is an interesting example of nonholonomic systems in control theory. There is a geometric condition equivalent to the rolling constraint which enables us to generalize the rolling motions for any two-dimensional Riemannian manifolds. This system has a five-dimensional phase space. In order to study the controllability of the rolling surfaces, we lift the system to a six-dimensional space and show that the lifted system is controllable unless the two surfaces have isometric universal covering spaces. In the non-controllable case there are some three-dimensional orbits each of which corresponds to an isometry of the universal covering spaces.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.