Christopher Croke

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On the generation of smooth three-dimensional rigid body motions. Abstract This paper addresses the problem of generating smooth trajectories between an initial and final position and orientation in space. The main idea is to define a functional depending on velocity or its derivatives that measures the smoothness of a trajectory and find trajectories that(More)
The set of rigid body motions forms the Lie group SE(3), the special Euclidean group in three dimensions. In this paper we investigate Riemannian metrics and aane connections on SE(3) that are suited for kinematic analysis and robot trajectory planning. In the rst part of the paper, we study metrics whose geodesics are screw motions. We prove that no(More)
1 Introduction The purpose of this chapter is to survey some recent results and state open questions concerning the rigidity of Riemannian manifolds. The starting point will be the boundary rigidity and conjugacy rigidity problems. These problems are connected to many other problems (Mostow-Margulis type rigidity, isopectral problems, isoperi-metric(More)
The set of spatial rigid body motions forms a Lie group known as the special Euclidean group in three dimensions, ¢ ¤ £ (3). Chasles's theorem states that there exists a screw motion between two arbitrary elements of ¢ ¤ £ (3). In this paper we investigate whether there exist a Riemannian metric whose geodesics are screw motions. We prove that no Riemannian(More)
We prove certain optimal systolic inequalities for a closed Riemannian manifold (X, g), depending on a pair of parameters , n and b. Here n is the dimension of X, while b is its first Betti number. The proof of the inequalities involves constructing Abel-Jacobi maps from X to its Jacobi torus T b , which are area-decreasing (on b-dimensional areas), with(More)
We prove the filling area conjecture in the hyperellip-tic case. In particular, we establish the conjecture for all genus 1 fillings of the circle, extending P. Pu's result in genus 0. We translate the problem into a question about closed ovalless real surfaces. The conjecture then results from a combination of two ingredients. On the one hand, we exploit(More)