Accelerated Hardware Using Dual Quaternions
Techniques for concatenating, interpolating and upsampling pose transforms represented as dual quaternions are described, including: (1) derivation of a complex-valued matrix form of dual quaternions and dual quaternion operations; (2) derivation of a transformation operator on position vectors which obviates an explicit conversion to a classical 4×4 spatial transformation matrix and keeps results in complex-valued matrix space; (3) design for a generic lookup table system for functions to supply logarithm and exponentiations of the dual quaternion in its native format with trigonometry lookup tables to avoid precision issues when denominators tend to zero; and (4) a mechanism for wrapping the complex-exponentiation together with a simple complex arithmetic unit for computing dual quaternion macro-operations in both native dual quaternion space and through simplifications of the equivalent complex-valued matrix to compute dual quaternion operations such as inverses, multiplications, logarithms and exponentials in order to chain the pose transformations encoded within.
1 . A device comprising:
a first interface that accepts at least one sample of a state of a chain of at least one rigid body transformation represented as a dual quatemion annotated with a first set of specific sampled points in time;
a second interface that accepts a second set containing at least one of the specific sampled points in time;
wherein an output of the device contains a rigid body transformation chain of the first interface resampled at the second set containing at least one of the specific sampled points in time.
2 . The device of claim 1 wherein each of the rigid body transform chains is associated with a subset of the first set of specific sampled points in time.
3 . The device of claim 2 wherein a parallel data pipeline is used to resample the dual quatemion transformations.
4 . The device of claim 2 wherein a scalar complex-valued data pipeline is used to resample the dual quatemion transformations.
5 . The device of claim 2 wherein at least one transformation is a chain provided by a tracking device synchronized to a different clock.
6 . A device comprising:
a first interface that accepts a chain of at least one rigid body transformations represented as dual quaternions;
a second interface that accepts at least one of three-dimensional vectors;
wherein an output of the device contains a set of the at least one of three-dimensional vectors having been transformed by a rigid body transformation chain of the first interface.
7 . The device of claim 6 wherein a parallel data pipeline is used to apply a transformation to at least one direction vector.
8 . The device of claim 6 wherein a parallel data pipeline is used to apply a transformation to at least one position vector.
9 . The device of claim 6 wherein a scalar complex-valued data pipeline is used to apply a transformation to at least one direction vector.
10 . The device of claim 6 wherein a scalar complex-valued data pipeline is used to apply a transformation to at least one position vector.
11 . A device comprising:
a first interface that accepts at least one sample of a state of a chain of at least one rigid body transformation represented as dual quaternions annotated with a first set of specific sampled points in time;
a second interface that delivers a synchronous stream of three-dimensional vector data, wherein at least one of the three-dimensional vectors is associated with a discrete synchronous point in time;
wherein an output of the device contains the synchronous stream of the second interface transformed by the at least one rigid body transformation chain of the first interface resampled at a time of the synchronous stream.
12 . The device of claim 11 wherein each transform in the chain is associated with a subset of a first set of specific sampled points in time.
13 . The device of claim 12 wherein at least one transformation chain is provided by a tracking device synchronized to a different clock.
14 . The device of claim 12 wherein a parallel data pipeline is used to apply the transformation to at least one direction vector.
15 . The device of claim 12 wherein a parallel data pipeline is used to apply the transformation to at least one position vector.
16 . The device of claim 12 wherein a scalar complex-valued data pipeline is used to apply the transformation to at least one direction vector.
17 . The device of claim 12 wherein a scalar complex-valued data pipeline is used to apply the transformation to at least one position vector.