F. Silva Leite

THE DYNAMIC INTERPOLATION PROBLEM: ON RIEMANNIAN MANIFOLDS, LIE GROUPS, AND SYMMETRIC SPACES

We consider the dynamic interpolation problem for nonlinear control systems modeled by second-order differential equations whose configuration space is a Riemannian manifoldM. In this problem we are given an ordered set of points inM and would like to generate a trajectory of the system through the application of suitable control functions, so that the resulting trajectory in configuration space interpolates the given set of points. We also impose smoothness constraints on the trajectory and typically ask that the trajectory be also optimal with respect to some physically interesting cost function. Here we are interested in the situation where the trajectory is twice continuously differentiable and the Lagrangian in the optimization problem is given by the norm squared acceleration along the trajectory. The special cases whereM is a connected and compact Lie group or a homogeneous symmetric space are studied in more detail.

The De Casteljau Algorithm on Lie Groups and Spheres

AbstractWe examine the De Casteljau algorithm in the context of Riemannian symmetric spaces. This algorithm, whose classical form is used to generate interpolating polynomials in

Elastic Curves as Solutions of Riemannian and Sub-Riemannian Control Problems

\mathbb{R}^n

Closed forms for the exponential mapping on matrix Lie groups based on Putzer’s method

, was also generalized to arbitrary Riemannian manifolds by others. However, the implementation of the generalized algorithm is difficult since detailed structure, such as boundary value expressions, has not been available. Lie groups are the most simple symmetric spaces, and for these spaces we develop expressions for the first and second order derivatives of curves of arbitrary order obtained from the algorithm. As an application of this theory we consider the problem of implementing the generalized De Casteljau algorithm on an m-dimensional sphere. We are able to fully develop the algorithm for cubic splines with Hermite boundary conditions and more general boundary conditions for arbitrary m.

Rolling Stiefel manifolds

Abstract We continue the work of Crouch and Silva Leite on the geometry of cubic polynomials on Riemannian manifolds. In particular, we generalize the theory of Jacobi fields and conjugate points and present necessary and sufficient optimality conditions

Riemannian Means as Solutions of Variational Problems

Abstract. We consider the nonlinear dynamic interpolation problem on Riemannian manifolds and, in particular, on connected and compact Lie groups. Basically we force the dynamic variables of a control system to pass through specific points in the configuration space, while minimizing a certain energy function, by a suitable choice of the controls. The energy function we consider depends on the velocity and acceleration along trajectories. The solution curves can be seen as generalizations of the classical splines in tension for the Euclidean space. The relations with sub-Riemannian optimal control problems are explained.

Computing the square root and logarithm of real P -orthogonal matrix

Abstract We study the orthogonal solutions of the matrix equation XJ − JX T = M , where J is symmetric positive definite and M is skew-symmetric. This equation arises in the discrete version of the dynamics of a rigid body, investigated by Moser and Veselov (Commun. Math. Phys. 139 (1991) 217). We show connections between orthogonal solutions of this equation and solutions of a certain algebraic Riccati equation. This will bring out the symplectic geometry of the Moser–Veselov equation and also reduces most computational issues about solutions to finding invariant subspaces of a certain Hamiltonian matrix. Necessary and sufficient conditions for the existence of orthogonal solutions (and methods to compute them) are presented. Our method is contrasted with the Moser–Veselov approach (Commun. Math. Phys. 139 (1991) 217). We also exhibit explicit solutions of a particular case of the Moser–Veselov equation, which appears associated with the continuous version of the dynamics of a rigid body.

Theoretical and numerical considerations about Padé approximants for the matrix logarithm

We present closed forms for the exponential of some infinitesimal generators of Lie groups which play an important role in physics and engineering applications. These explicit forms are based on the Putzer’s method. We also compare this methodology and results with related work by other authors.