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- Milos Zefran, Vijay Kumar, Christopher Croke
- IEEE Trans. Robotics and Automation
- 1998

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)

We present a theorem of Lancret for general helices in a 3-dimensional real-space-form which gives a relevant difference between hyperbolic and spherical geometries. Then we study two classical problems for general helices in the 3-sphere: the problem of solving natural equations and the closed curve problem.

- C Croke, V Schroeder, +6 authors Viktor Schroeder
- 2010

The fundamental group of compact manifolds without conjugate points.

We consider compact Riemannian manifolds (M,∂M, g) with boundary ∂M and metric g on which a finite group Γ acts freely. We determine the extent to which certain rigidity properties of (M,∂M, g) descend to the quotient (M/Γ, ∂/Γ, g). In particular, we show by example that if (M,∂M, g) is boundary rigid then (M/Γ, ∂/Γ, g) need not be. On the other hand, lens… (More)

Let 1 → H → G → Z → 1 be an exact sequence of hyperbolic groups induced by an automorphism φ of the free group H. Let H1(⊂ H) be a finitely generated distorted subgroup of G. Then there exist N > 0 and a free factor K of H such that the conjugacy class of K is preserved by φN and H1 contains a finite index subgroup of a conjugate of K. This is an analog of… (More)

The lines of curvature on a cyclide of Dupin are circular arcs. A surface which carries two orthogonal families of circular arcs must arise as an integral surface of an overdetermined exterior differential system. We show that the only solutions of this system are the cyclides of Dupin.

Let (M, g) be a compact Riemannian manifold with universal covering M̃ . The simplest asymptotic invariant which can be associated to (M, g) is the exponential growth rate h(g) of volume on the universal covering, called the volume entropy. If B r (p) denotes the geodesic ball of radius r about p ∈ M̃ and volg(B r (p)) describes its volume with respect to… (More)

It is shown that a collapsible, compact, connected, simplicial polyhedron admits a cubical subdivision and a median convexity, such that all cubes are convex subspaces with a convexity of subcubes. Conversely, a compact, connected, cubical polyhedron with a convexity as described admits a collapsible simplicial subdivision. Such a convexity, when it exists,… (More)

- Milos Zefran, Vijay Kumar, Christopher Croke
- I. J. Robotics Res.
- 1999

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)

Let Ω be an (n + 1)-dimensional compact Riemannian manifold with nonnegative Ricci curvature and nonempty boundary M = ∂Ω. Assume that the principal curvatures of M are bounded from below by a positive constant c. In this paper, we prove that the first nonzero eigenvalue λ1 of the Laplacian of M acting on functions on M satisfies λ1 ≥ nc2 with equality… (More)