Smooth invariant interpolation of rotations

@article{Park1997SmoothII,
  title={Smooth invariant interpolation of rotations},
  author={F. Park and B. Ravani},
  journal={ACM Trans. Graph.},
  year={1997},
  volume={16},
  pages={277-295}
}
We present an algorithm for generating a twice-differentiable curve on the rotation group SO(3) that interpolated a given ordered set of rotation matrices at their specified knot times. In our approach we regard SO(3) as a Lie group with a bi-invariant Riemannian metriac, and apply the coordinate-invariant methods of Riemannian geometry. The resulting rotation curve is easy to compute, invariant with respect to fixed and moving reference frames, and also approximately minimizes angular… Expand
Cubic spline algorithms for orientation interpolation
This article presents a class of spline algorithms for generating orientation trajectories that approximately minimize angular acceleration. Each algorithm constructs a twice-differentiable curve onExpand
Robust Rotation Interpolation Based on SO(n) Geodesic Distance
A novel interpolation algorithm for smoothing of successive rotation matrices based on the geodesic distance of special orthogonal group SO(n) is proposed. The derived theory is capable of achievingExpand
A Constructive Approximation of Interpolating Bézier Curves on Riemannian Symmetric Spaces
TLDR
A new method to approximate curves that interpolate a given set of time-labeled data on Riemannian symmetric spaces is proposed and it is proved that the approximates enjoy a number of nice properties. Expand
Closed Form solution for C 2 Orientation interpolation
TLDR
This work proposes using C 2 interpolatory (cardinal) basis for C 2 smooth quaternion interpolation problem, which outperforms all alternatives and, being explicit, is absolutely stable. Expand
Higher order geodesics in Lie groups
  • Tomasz Popiel
  • Mathematics, Computer Science
  • Math. Control. Signals Syst.
  • 2007
TLDR
A duality theory, based on the invariance of the Euler–Lagrange equation under group inversion, is developed and the solution is presented in the case of the rotation group SO(3), which is important in rigid body motion planning. Expand
Representing Rotations and Orientations in Geometric Computing
  • Jehee Lee
  • Computer Science, Medicine
  • IEEE Computer Graphics and Applications
  • 2008
TLDR
Co Coordinate-free geometric programming and affine geometry, which makes a distinction between points and vectors and defines operations for combining them, inspires this approach. Expand
Generation Method of Bezier Curves and Surfaces on Lie Groups
TLDR
The goal of this paper is to generalize the concept of Bezier curves and surfaces defined on the vector space to Lie groups, which is a new generation method of curves (called BeZier curves) on Lie groups to smooth motion interpolation or smooth trajectory generation for moving rigid body in space. Expand
Bézier curves and C2 interpolation in Riemannian manifolds
TLDR
This work compute the endpoint velocities and (covariant) accelerations of a generalised Bezier curve of arbitrary degree and use the formulae to express the curve's control points in terms of these quantities, and shows that C^2 continuity is equivalent to a simple relationship, involving the global symmetries at knot points, between the control points of neighbouring curve segments. Expand
Duality and Riemannian cubics
  • L. Noakes
  • Mathematics, Computer Science
  • Adv. Comput. Math.
  • 2006
TLDR
Results of the present paper include explicit solutions of the linking equation by quadrature in terms of the Lie quadratic, when G is SO(3) or SO(1,2); results on asymptotics of x follow from known properties of null Lie quadratics. Expand
Applications of Conformal Geometric Algebra in Computer Vision and Graphics
TLDR
A new method for pose and position interpolation based on CGA is discussed which firstly allows for existing interpolation methods to be cleanly extended to pose andposition interpolation, but also allows for this to be extended to higher-dimension spaces and all conformal transforms (including dilations). Expand
...
1
2
3
4
5
...

References

SHOWING 1-10 OF 26 REFERENCES
Smooth interpolation of orientations with angular velocity constraints using quaternions
TLDR
Methods to smoothly interpolate orientations, given N rotational key frames of an object along a trajectory, are presented and the user is allowed to impose constraints on the rotational path, such as the angular velocity at the endpoints of the trajectory. Expand
Visualization of moving objects using dual quaternion curves
  • B. Jüttler
  • Mathematics, Computer Science
  • Comput. Graph.
  • 1994
TLDR
In order to apply the powerful methods of computer-aided geometric design, an interpolating motion whose trajectories are rational Bezier curves is constructed. Expand
Planning of smooth motions on SE(3)
  • M. Žefran, Vijay R. Kumar
  • Mathematics, Computer Science
  • Proceedings of IEEE International Conference on Robotics and Automation
  • 1996
TLDR
The authors derive necessary conditions for the shortest distance and minimum jerk trajectories and solve the resulting two-point boundary value problem. Expand
Bézier Curves on Riemannian Manifolds and Lie Groups with Kinematics Applications
In this article we generalize the concept of Bezier curves to curved spaces, and illustrate this generalization with an application in kinematics. We show how De Casteljau's algorithm forExpand
Elastic curves on the sphere
TLDR
In this paper, equations for the main invariants of spherical elastic curves are given and a new method for solving geometrically constraint differential equations is used to compute the curves for given initial values. Expand
Animating rotation with quaternion curves
TLDR
A new kind of spline curve is presented, created on a sphere, suitable for smoothly in-betweening (i.e. interpolating) sequences of arbitrary rotations, without quirks found in earlier methods. Expand
Theory of Lie Groups (PMS-8)
This famous book was the first treatise on Lie groups in which a modern point of view was adopted systematically, namely, that a continuous group can be regarded as a global object. To develop thisExpand
Animating rotation with quaternion curves
TLDR
The rotations of solid bodies roll and tumble through space are best described using a four coordinate system, quaternions, as is shown in this animation. Expand
Hermite Interpolation of Solid Orientations with Circular Blending Quaternion Curves
Construction methods are presented that generate Hermite interpolation quaternion curves on SO(3). Two circular curves C1(t) and C2(t), 0 ≤ t ≤ 1, are generated that interpolate two orientations q1Expand
On Smooth Interpolation
For a given set of n distinct points (x1, y1), . . . , (xn, yn) and a given system of continuously differentiable functions U = {ui} ∞ i=0, u0 ≡ 1, such that the system of the derivatives V = {vi} ∞Expand
...
1
2
3
...