Many useful identities pertaining to quaternion multiplications are generalized. Among them multiplicative commutativity is the most powerful. Since quaternion space includes the 3D vector space, the physical quantities related to rotations, such as angular displacement, velocity, acceleration, and momentum, are shown to be vector quaternions, and their expressions in quaternion space are derived. These kinematic and dynamic differential equations are further shown to be invertible due to the fact that they are written in quaternion space, and the highest order term of the rotation parameters can be expressed explicitly in closed form. >

Lectures on quaternions

Quaternion kinematic and dynamic differential equations

The Kalman filter provides a powerful mathematical framework within which a minimum mean-square-error estimate of a user's position and orientation can be tracked using a sequence of single sensor observations, as opposed to groups of observations. We refer to this new approach as single-constraint-at-a-time or SCAAT tracking. The method improves accuracy by properly assimilating sequential observations, filtering sensor measurements, and by concurrently autocalibrating mechanical or electrical devices. The method facilitates user motion prediction, multisensor data fusion, and in systems where the observations are only available sequentially it provides estimates at a higher rate and with lower latency than a multiple-constraint approach.
Improved accuracy is realized primarily for three reasons. First, the method avoids mathematically treating truly sequential observations as if they were simultaneous. Second, because each estimate is based on the observation of an individual device, perceived error (statistically unusual estimates) can be more directly attributed to the corresponding device. This can be used for concurrent autocalibration which can be elegantly incorporated into the existing Kalman filter. Third, the Kalman filter inherently addresses the effects of noisy device measurements. Beyond accuracy, the method nicely facilitates motion prediction because the Kalman filter already incorporates a model of the user's dynamics, and because it provides smoothed estimates of the user state, including potentially unmeasured elements. Finally, in systems where the observations are only available sequentially, the method can be used to weave together information from individual devices in a very flexible manner, producing a new estimate as soon as each individual observation becomes available, thus facilitating multisensor data fusion and improving the estimate rates and latencies.
The most significant aspect of this work is the introduction and exploration of the SCAAT approach to 3D tracking for virtual environments. However I also believe that this work may prove to be of interest to the larger scientific and engineering community in addressing a more general class of tracking and estimation problems.

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SCAAT: incremental tracking with incomplete information

Systems and methods according to the present invention describe 3D pointing devices which enhance usability by transforming sensed motion data from a first frame of reference (e.g., the body of the 3D pointing device) into a second frame of reference (e.g., a user's frame of reference). One exemplary embodiment of the present invention removes effects associated with a tilt orientation in which the 3D pointing device is held by a user.

3D pointing devices with orientation compensation and improved usability

This intermediate-level tutorial provides a comprehensive approach to the visualization of quaternions and their relationships to computer graphics and scientific visualization. The introduction focuses on a selection of everyday phenomena involving rotating objects whose explanation for an audience that is technically trained but not pure mathematicians is essentially impossible without a quaternion visualization. The course will then pursue selected examples of quaternion-based visualization methods to help explain the behavior of quaternion manifolds: quaternion representations of orientation frames attached to curves, surfaces, and volumes.

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Visualizing Quaternions

The main topics of this technical report are quaternions, their mathematical properties, and how they can be used to rotate objects. We introduce quaternion mathematics and discuss why quaternions are a better choice for implementing rotation than the well-known matrix implementations. We then treat di erent methods for interpolation between series of rotations. During this treatment we give complete proofs for the correctness of the important interpolation methods Slerp and Squad . Inspired by our treatment of the di erent interpolation methods we develop our own interpolation method called Spring based on a set of objective constraints for an optimal interpolation curve. This results in a set of di erential equations, whose analytical solution meets these constraints. Unfortunately, the set of di erential equations cannot be solved analytically. As an alternative we propose a numerical solution for the di erential equations. The di erent interpolation methods are visualized and commented. Finally we provide a thorough comparison of the two most convincing methods (Spring and Squad). Thereby, this report provides a comprehensive treatment of quaternions, rotation with quaternions, and interpolation curves for series of rotations.

/pdf/quaternions-interpolation-and-animation-3d5ypnjmv2.pdf

Lectures on quaternions

Citations

Quaternion kinematic and dynamic differential equations

SCAAT: incremental tracking with incomplete information

3D pointing devices with orientation compensation and improved usability

Visualizing Quaternions

Quaternions, Interpolation and Animation

Related Papers (5)

Elements of Quaternions

Quaternions and rotation sequences : a primer with applications to orbits, aerospace, and virtual reality

Animating rotation with quaternion curves

Rotations, quaternions, and double groups

Classical Mechanics