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.

We present a procedure to generate smooth interpolating curves on submanifolds, which are given in closed form in terms of the coordinates of the embedding space. In contrast to other existing methods, this approach makes the corresponding algorithm easy to implement. The idea is to project the prescribed data on the manifold onto the affine tangent space at a particular point, solve the interpolation problem on this affine subspace, and then project the resulting curve back on the manifold. One of the novelties of this approach is the use of rolling mappings. The manifold is required to roll on the affine subspace like a rigid body, so that the motion is described by the action of the Euclidean group on the embedding space. The interpolation problem requires a combination of a pullback/push forward with rolling and unrolling. The rolling procedure by itself highlights interesting properties and gives rise to a new, but simple, concept of geometric polynomial curves on manifolds. This paper is an extension of our previous work, where mainly the 2-sphere case was studied in detail. The present paper includes results for the n-sphere, orthogonal group SO n, and real Grassmann manifolds. In particular, we present the kinematic equations for rolling these manifolds along curves without slip or twist, and derive from them formulas for the parallel transport of vectors along curves on the manifold.

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In the past few years there has been a growing interest on geometric frameworks to learn supervised classification models on Riemannian manifolds [31,27]. A popular framework, valid over any Riemannian manifold, was proposed in [31] for binary classification. Once moving from binary to multi-class classification this paradigm is not valid anymore, due to the spread of multiple positive classes on the manifold [27]. It is then natural to ask whether the multi-class paradigm could be extended to operate on a large class of Riemannian manifolds. We propose a mathematically well-founded classification paradigm that allows to extend the work in [31] to multi-class models, taking into account the structure of the space. The idea is to project all the data from the manifold onto an affine tangent space at a particular point. To mitigate the distortion induced by local diffeomorphisms, we introduce for the first time in the computer vision community a well-founded mathematical concept, so-called Rolling map [21,16]. The novelty in this alternate school of thought is that the manifold will be firstly rolled (without slipping or twisting) as a rigid body, then the given data is unwrapped onto the affine tangent space, where the classification is performed.

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