Fátima Silva Leite

Smooth Trajectory Planning for Fully Automated Passengers Vehicles: Spline and Clothoid Based Methods and Its Simulation

A new approach for mobility, providing an alternative to the private passenger car, by offering the same flexibility but with much less nuisances, is emerging, based on fully automated electric vehicles. A fleet of such vehicles might be an important element in a novel individual, door-to-door, transportation system to the city of tomorrow. For fully automated operation, trajectory planning methods that produce smooth trajectories, with low associated accelerations and jerk, for providing passenger’s comfort, are required. This chapter addresses this problem proposing an approach that consists of introducing a velocity planning stage to generate adequate time sequences for usage in the interpolating curve planners. Moreover, the generated speed profile can be merged into the trajectory for usage in trajectory-tracking tasks like it is described in this chapter, or it can be used separately (from the generated 2D curve) for usage in path-following tasks. Three trajectory planning methods, aided by the speed profile planning, are analysed from the point of view of passengers’ comfort, implementation easiness, and trajectory tracking.

Rolling Riemannian Manifolds to Solve the Multi-class Classification Problem

In the past few years there has been a growing interest on geometric frameworks to learn supervised classification models on Riemannian manifolds [32, 28]. A popular framework, valid over any Riemannian manifold, was proposed in [32] 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 [28]. 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 [32] 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 [22, 17]. 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.

A new geometric algorithm to generate smooth interpolating curves on Riemannian manifolds

This paper presents a new geometric algorithm to construct a C smooth spline curve that interpolates a given set of data (points and velocities) on a complete Riemannian manifold. Although based on a modification of the de Casteljau procedure, our algorithm is implemented in three steps only independently of the required degree of smoothness, and therefore introduces a significant reduction in complexity. The key role is played by the choice of an appropriate smoothing function which is defined as soon as the degree of smoothness is fixed.

Kinematics for rolling a Lorentzian sphere

We derive the equations of motion for the n-dimensional Lorentzian sphere (one-sheet hyperboloid) rolling, without slipping and twisting, over the affine tangent space at a point. Both manifolds are endowed with semi-Riemannian metrics, induced by the Lorentzian metric on the embedding manifold which is the generalized Minkowski space. The kinematic equations turn out to be a nonlinear control system evolving on a connected subgroup of the Poincaré group. The controls correspond to the choice of the curves along which the Lorentzian sphere rolls. Controllability of this rolling system will be proved by showing that the corresponding distribution is bracket-generating.

Hamiltonian Structure of Generalized Cubic Polynomials

Abstract We present a Hamiltonian formulation of a second order variational problem on a Riemannian manifold (Q, ), which gives rise to generalized cubic polynomials on Q, and explore the possibility of writing down the extremal solutions of that problem as a flow in the space ∪ q ϵ Q T q Q ⊕ T q * Q ⊕ T q * Q . For that we utilize the connection ∇ on Q, corresponding to the metric . We exhibit the extremal equations in Hamiltonian form and identify the correct symplectic form. In general the results depend upon a choice of frame for TQ, but for the special situation when Q is a Lie group G with Lie algebra G, our results are global and the flow reduces to a flow on G × g ×g* ×g*.

Second order optimality conditions for a higher order variational problem on a Riemannian manifold

In this paper, we derive second order optimality conditions for a higher order variational problem on a general Riemannian manifold, which can be viewed as an extension of the minimizing acceleration problem in Euclidean space and yields the geometric generalization of the classical cubic polynomials. This continues the work initiated by Crouch and Silva Leite (1995). In particular, we define conjugate points and prove a necessary and sufficient condition for optimality, in the absence of such singularities.

State Estimation for Systems on SE(3) with Implicit Outputs: An Application to Visual Servoing

Abstract Motivated by applications in visual servoing, we consider the state estimation problem for a class of systems described by implicit outputs and whose state lives in the special Euclidean group SE(3). We propose an observer in the group of motion SE(3) that preserves invariance and therefore takes explicitly into consideration the geometry of the problem. We discuss conditions under which the linearized state estimation error converges exponentially fast. Furthermore, we analyze the problem when the system is subject to disturbances and noises and show that the estimate converges to a neighborhood of the real solution. The size of the neighborhood increases/decreases gracefully with the bound of the disturbance and noise. We apply and illustrate these results through an application of position and attitude estimation of a rigid body using measurements from a camera attached to the rigid body.

An algorithm based on rolling to generate smooth interpolating curves on ellipsoids

We present an algorithm to generate a smooth curve interpolating a set of data on an

An Exponential Observer for Systems on SE.3/ with Implicit Outputs

n

Rolling motions of pseudo-orthogonal groups

-dimensional ellipsoid, which is given in closed form. This is inspired by an algorithm based on a rolling and wrapping technique, described in [11] for data on a general manifold embedded in Euclidean space. Since the ellipsoid can be embedded in an Euclidean space, this algorithm can be implemented, at least theoretically. However, one of the basic steps of that algorithm consists in rolling the ellipsoid, over its affine tangent space at a point, along a curve. This would allow to project data from the ellipsoid to a space where interpolation problems can be easily solved. However, even if one chooses to roll along a geodesic, the fact that explicit forms for Euclidean geodesics on the ellipsoid are not known, would be a major obstacle to implement the rolling part of the algorithm. To overcome this problem and achieve our goal, we embed the ellipsoid and its affine tangent space in