• Corpus ID: 7075201

Quaternions and Rotations *

  title={Quaternions and Rotations *},
  author={Yan-Bin Jia},
Up until now we have learned that a rotation in R3 about an axis through the origin can be represented by a 3×3 orthogonal matrix with determinant 1. However, the matrix representation seems redundant because only four of its nine elements are independent. Also the geometric interpretation of such a matrix is not clear until we carry out several steps of calculation to extract the rotation axis and angle. Furthermore, to compose two rotations, we need to compute the product of the two… 

Figures from this paper

Orientation Modeling Using Quaternions and Rational Trigonometry

This research presents a new procedure for model orientation using rational trigonometry and quaternion notation, avoiding trigonometric functions for calculations, and aims to compare the efficiency of a rational implementation to classical modeling using the techniques mentioned.

Helmert Transformation Problem. From Euler Angles Method to Quaternion Algebra

The three-dimensional coordinate’s transformation from one system to another, and more specifically, the Helmert transformation problem, is one of the most well-known transformations in the field of engineering and its solution was investigated for specific data using three different methods.

An Algorithm for Quaternion–Based 3D Rotation

A new algorithm for quaternion-based spatial rotation is presented which reduces the number of underlying real multiplications, and the proposed algorithm can compute the same result in only 14 real multiplier cases.

Survey of Higher Order Rigid Body Motion Interpolation Methods for Keyframe Animation and Continuous-Time Trajectory Estimation

This survey carefully analyze the characteristics of higher order rigid body motion interpolation methods to obtain a continuous trajectory from a discrete set of poses and concludes that split interpolation in R3 × SU(2) is preferable for most applications.

On the Orientation Planning with Constrained Angular Velocity and Acceleration at Endpoints

To impose the unitariness condition critically required for quaternion representation of orientation, an on-line update mechanism is developed which successively reparameterizes the polynomials constructing the spline, towards suppressing distortions that the normalization operation might incur.

A Pose-Graph Optimization tool for MATLAB

This paper presents the development and implementation of a pose-graph optimization tool for MATLAB that consists in generating a graph from the poses of the robot and from the constraints of measurements between poses, followed by the optimization of this graph to obtain a consistent trajectory.

Distributed Dual Quaternion Based Localization of Visual Sensor Networks

An estimation scheme that exploits the unit dual quaternion algebra to describe the sensors pose is proposed that is beneficial in the formulation of the optimization scheme allowing to solve the localization problem without designing two interlaced position and orientation estimators, thus improving the estimation error distribution over the two pose components and the overall localization performance.

Building up an Inertial Navigation System Using Standard Mobile Devices

This paper presents the development of a PNS (Pedestrian Navigation System), which utilizes accelerometer, gyroscope and magnetometer data to enable accurate positioning and combines the most promising techniques and describes improvements.

Inscribed Tverberg‐type partitions for orbit polytopes

Tverberg's theorem states that any set of t(r,d)=(r−1)(d+1)+1$t(r,d)=(r-1)(d+1)+1$ points in Rd$\mathbb {R}^d$ can be partitioned into r subsets whose convex hulls have non‐empty r‐fold intersection.



A novel Quaternion integration approach for describing the behaviour of non-spherical particles

There are three main frameworks to describe the orientation and rotation of non-spherical particles: Euler angles, rotation matrices and unit Quaternions. Of these methods, the latter seems the most

Quaternions, Interpolation and Animation

This report provides a comprehensive treatment of quaternion mathematics, rotation with quaternions, and interpolation curves for series of rotations with a thorough comparison of the two most convincing methods.

Closed-form solution of absolute orientation using unit quaternions

A closed-form solution to the least-squares problem for three or more paints is presented, simplified by use of unit quaternions to represent rotation.

Animating rotation with quaternion curves

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.

Quaternions and Rotation Sequences

In this paper we introduce and define the quaternion; we give a brief introduction to its properties and algebra, and we show, what appears to be, its primary application — the quaternion rotation

A Method for Registration of 3-D Shapes

A general-purpose, representation-independent method for the accurate and computationally efficient registration of 3-D shapes including free-form curves and surfaces based on the iterative closest point (ICP) algorithm.

The Representation, Recognition, and Locating of 3-D Objects

This work proposes the paradigm of recognizing objects while locating them as a prediction and verifi cation scheme that makes efficient use of the shape representation and the matching algorithm, which are general and can be used for other types of data, such as ultrasound, stereo, and tactile.

Identification of Partially Obscured Objects in Two and Three Dimensions by Matching Noisy Characteristic Curves

Efficient methods for smoothing the noisy data and for matching portions of the observed object boundaries (or of characteristic curves lying on bounding surfaces of 3-D objects) to prestored models are described.

Numerical Recipes in C, 2nd Edition