Finally, we achieve the following algorithm for computing the http://www.selleckchem.com/products/baricitinib-ly3009104.html Riemannian mean P�� on SE(n).Algorithm 5 ��Given N matrices Pk, k = 1,2,��, N, on SE(n), their Riemannian mean P�� is computed by the following iterative method.Store (1/N)��k=1Nbk to b��.Set A��=A1 as an initial input, and choose a desired tolerance �� > 0.If ||��k=1Nlog?(A��TAk)||F<��, then stop.Otherwise, update A��=A��exp?-�š�k=1Nlog?(A��TAk), and go to step (3).3.3. Simulations on SE(3)Let us consider a rigid object W in the Euclidean space undergoing a rigid body Euclidean motion SE(3). Suppose that the coordinate of the center of gravity in W is dW 3; then, the optimal trajectory from the configuration P to Q is the curve D(t) such that(D(t)1)=��P,Q(t)(dW1),(40)where t [0,1] and ��P,Q(t) denotes the geodesic connecting P and Q on SE(3)(see Figure 1).

For the configuration of two points P and Q, as shown in Figure 2, given by the angular velocity ��P,��Q of the rigid body and the linear velocity vP, vQ, we choose ��P = (��/2)(0,1, 1), vP = (0,0, 0),��Q = ��(1/4,0, ?1/2), and vQ = (4.380, ?1.348,3.690); then, we obtain their Riemannian mean according to Algorithm 5, which is just the middle point PQ from (24).Figure 1The rigid motion D(t) from P to Q.Figure 2 The Riemannian mean PQ.4. The Riemannian Mean on UP(n)In this section, the Riemannian mean of N given points on the unipotent matrix group UP(n) is considered. UP(n) is a noncompact matrix Lie group as well. Moreover, in the special case n = 3, it is the Heisenberg group H(3).4.1.

About UP(n)The set of all of the uppertriangular n �� n matrices with diagonal elements that are all one is called unipotent matrices group, denoted by UP(n).In fact, given an invertible matrix C UP(n), there is a neighborhood U of C such that every matrix in U is also in UP(n), so UP(n) is an open subset of n��n. Furthermore, the matrix product P ? Q is clearly a smooth function of Drug_discovery the entries of P and Q, and P?1 is a smooth function of the entries of P. Thus, UP(n) is a Lie group. On the other hand, it can be verified that UP(n) is of dimension n(n ? 1)/2 and is nilpotent. Since we can use the nonzero elements Cij,i < j, directly as global coordinate functions for UP(n), the manifold underlying UP(n) is diffeomorphic to n(n?1)/2. Therefore, UP(n) is not compact, but simply connected.The Lie algebra (n) of UP(n) consists of uppertriangular matrices T with diagonal elements Tii = 0, i = 1,��, n. It is an indispensable tool which gives a realization of the Heisenberg commutation relations of quantum mechanics in the 3-dimensional case [17].Moreover, it is the fact that both C ? I and T are all nilpotent matrices, for any C UP(n) and T (n).