Margarida Camarinha

Bio: Margarida Camarinha is an academic researcher from University of Coimbra. The author has contributed to research in topics: Lie group & Obstacle avoidance. The author has an hindex of 8, co-authored 31 publications receiving 360 citations.

On the geometry of Riemannian cubic polynomials

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

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

Dynamic interpolation for obstacle avoidance on Riemannian manifolds

Abstract: This work is devoted to studying dynamic interpolation for obstacle avoidance. This is a problem that consists of minimising a suitable energy functional among a set of admissible curves subject to...

Dynamic interpolation for obstacle avoidance on Riemannian manifolds

Abstract: This work is devoted to studying dynamic interpolation for obstacle avoidance. This is a problem that consists of minimizing a suitable energy functional among a set of admissible curves subject to some interpolation conditions. The given energy functional depends on velocity, covariant acceleration and on artificial potential functions used for avoiding obstacles. We derive first-order necessary conditions for optimality in the proposed problem; that is, given interpolation and boundary conditions we find the set of differential equations describing the evolution of a curve that satisfies the prescribed boundary values, interpolates the given points and is an extremal for the energy functional. We study the problem in different settings including a general one on a Riemannian manifold and a more specific one on a Lie group endowed with a left-invariant metric. We also consider a sub-Riemannian problem. We illustrate the results with examples of rigid bodies, both planar and spatial, and underactuated vehicles including a unicycle and an underactuated unmanned vehicle.

Geometric control of mechanical systems

Abstract: This talk will outline a comprehensive set of modeling, analysis and design techniques for a class of mechanical systems. We concern ourselves with simple mechanical control systems, that is, systems whose Lagrangian is kinetic energy minus potential energy. Example devices include robotic manipulators, aerospace and underwater vehicles, and mechanisms that locomote exploiting nonholonomic constraints. Borrowing techniques from nonlinear control and geometric mechanics, we propose a coordinateinvariant control theory for this class of systems. First, we take a Riemannian geometric approach to modeling systems dened on smooth manifolds, subject to nonholonomic constraints, external forces and control forces. We also model mechanical systems on groups and symmetries. Second, we analyze some control-theoretic properties of this class of systems, including controllability, averaged response to oscillatory controls, and kinematic reductions. Finally, we exploit the modeling and analysis results to tackle control design problems. Starting from controllability and kinematic reduction assumptions we propose some algorithms for generating and tracking trajectories.

An intrinsic observer for a class of Lagrangian systems

Abstract: We propose a new design method of asymptotic observers for a class of nonlinear mechanical systems: Lagrangian systems with configuration (position) measurements. Our main contribution is to introduce a state (position and velocity) observer that is invariant under any changes of the configuration coordinates. The observer dynamics equations, as the Euler-Lagrange equations, are intrinsic. The design method uses the Riemannian structure defined by the kinetic energy on the configuration manifold. The local convergence is proved by showing that the Jacobian of the observer dynamics is negative definite (contraction) for a particular metric defined on the state-space, a metric derived from the kinetic energy and the observer gains. From a practical point of view, such intrinsic observers can be approximated, when the estimated configuration is close to the true one, by an explicit set of differential equations involving the Riemannian curvature tensor. These equations can be automatically generated via symbolic differentiations of the metric and potential up to order two. Numerical simulations for the ball and beam system, an example where the scalar curvature is always negative, show the effectiveness of such approximation when the measured positions are noisy or include high frequency neglected dynamics.

On the generation of smooth three-dimensional rigid body motions

Abstract: This paper addresses the problem of generating smooth trajectories between an initial and a final position and orientation in space. The main idea is to define a functional depending on velocity or its derivatives that measures smoothness of trajectories and find a trajectory that minimizes this functional. In order to ensure that the computed trajectories are independent of the parametrization of positions and orientations, we use the notions of Riemannian metric and covariant derivative from differential geometry and formulate the problem as a variational problem on the Lie group of spatial rigid body displacements. We show that by choosing an appropriate measure of smoothness, the trajectories can be made to satisfy boundary conditions on the velocities or higher order derivatives. Dynamically smooth trajectories can be obtained by incorporating the inertia of the system into the definition of the Riemannian metric. We state the necessary conditions for the shortest distance, minimum acceleration and minimum jerk trajectories.

Energy-minimizing splines in manifolds

Abstract: Variational interpolation in curved geometries has many applications, so there has always been demand for geometrically meaningful and efficiently computable splines in manifolds. We extend the definition of the familiar cubic spline curves and splines in tension, and we show how to compute these on parametric surfaces, level sets, triangle meshes, and point samples of surfaces. This list is more comprehensive than it looks, because it includes variational motion design for animation, and allows the treatment of obstacles via barrier surfaces. All these instances of the general concept are handled by the same geometric optimization algorithm, which minimizes an energy of curves on surfaces of arbitrary dimension and codimension.

Discrete Geometric Optimal Control on Lie Groups

Abstract: We consider the optimal control of mechanical systems on Lie groups and develop numerical methods that exploit the structure of the state space and preserve the system motion invariants. Our approach is based on a coordinate-free variational discretization of the dynamics that leads to structure-preserving discrete equations of motion. We construct necessary conditions for optimal trajectories that correspond to discrete geodesics of a higher order system and develop numerical methods for their computation. The resulting algorithms are simple to implement and converge to a solution in very few iterations. A general software implementation is provided and applied to two example systems: an underactuated boat and a satellite with thrusters.