The dynamic interpolation problem: On Riemannian manifolds, Lie groups, and symmetric spaces

Abstract

ABSTRACT. We consider the dynamic interpolation problem for nonlinear control systems modeled by second-order differential equations whose configuration space is a Riem~nnlan manifold M. In this problem we are given an ordered set of points in M 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 differentlable and the Lagrangian in the optimization problem is given by the norm squared acceleration along the trajectory. The special cases where M is a connected and compact Lie group or a homogeneous symmetric space are studied in more detail.

Fast Interpolation and Time-Optimization on Implicit Contact Submanifolds

“…Smooth interpolation on manifolds is well studied in the case where the manifold is equipped with a geodesic (e.g., SO(3) [5]). For many-DOF robots under arbitrary contact constraints, geodesics are difficult to derive.…”

“…Hermite interpolation on M is performed using the classic de Casteljau construction of a Bezier curve [5]. Given end points x 0 , x 1 ∈ M and tangents v 0 ∈ T x0 M, v 1 ∈ T x1 M, an interpolating curve is constructed first by calculating the Bezier control points:…”

Algebraic integrability for minimum energy curves

“…Under this assumption, the resulting trajectories are geometric cubic polynomials on the configuration space of the vehicle. These curves, which are generalizations to Riemannian manifolds of the classical and well established cubic polynomials on Euclidean spaces, have been first introduced by Noakes et al in [7] and further developed, for instance, in [1] and [2]. These optimization problems are formulated via a variational approach and the corresponding Euler-Lagrange equations have been derived in the general context of manifolds.…”

Inexact trajectory planning and inverse problems in the Hamilton–Pontryagin framework

“…As we will see in examples, a natural choice of Lagrangian leads to Riemannian cubics and their higher-order generalizations. This class of curves was introduced in Noakes et al [8] and has since been studied in a series of papers including [9][10][11][12][13][14]. Riemannian cubics appear in a variety of applications, for example, in the quantum control problem mentioned above, but also in computer graphics, robotics and spacecraft control [15][16][17][18].…”