Rational motion

In kinematics, the motion of a rigid body is defined as a continuous set of displacements. One-parameter motions can be definedas a continuous displacement of moving object with respect to a fixed frame in Euclidean three-space ("E"³), where the displacement depends on one parameter, mostly identified as time.

Rational Motions are defined by rational functions (ratio of two polynomial functions) of time. They produce rational trajectories, and therefore they integrate well with the existing NURBS (Non-Uniform Rational B-Spline) based industry standard CAD/CAM systems. They are readily amenable to the applications of existing Computer Aided Geometric Design (CAGD) algorithms. By combining kinematics of rigid body motions with NURBS geometry of curves and surfaces, methods have been developed for computer aided design of rational motions.

These CAD methods for motion design find applications in animation in computer graphics (key frame interpolation), trajectory planning in robotics (taught-position interpolation), spatial navigation in virtual reality, computer aided geometric design of motion via interactive interpolation, CNC tool path planning, and task specification in mechanism synthesis.

Background

There has been a great deal of research in applying the principles of Computer Aided Geometric Design (CAGD) to the problem of computer aided motion design. In recent years, it has been well established that rational Bezier and rational B-spline based curve representation schemes can be combined with dual quaternion representation Citation
author = McCarthy, J. M.
year = 1990
publisher = MIT Press Cambridge, MA, USA] of spatial displacements to obtain rational Bezier and B-splinemotions. Ge and Ravani cite journal
author = Ge, Q. J.; Ravani, B.
year = 1994
title = Computer Aided Geometric Design of Motion Interpolants
journal = Journal of mechanical design(1990)
volume = 116
issue = 3
pages = 756–762
doi = 10.1115/1.2919447
url = http://scitation.aip.org/getabs/servlet/GetabsServlet?prog=normal&id=JMDEDB000116000003000756000001&idtype=cvips&gifs=yes] , cite journal
author = Ge, Q. J.; Ravani, B.
year = 1994
title = Geometric Construction of Bézier Motions
journal = Journal of mechanical design(1990)
volume = 116
issue = 3
pages = 749–755
doi = 10.1115/1.2919446
url = http://scitation.aip.org/getabs/servlet/GetabsServlet?prog=normal&id=JMDEDB000116000003000749000001&idtype=cvips&gifs=yes] developed a new framework for geometric constructionsof spatial motions by combining the concepts from kinematics and CAGD. Their work was built upon the seminal paper of Shoemake cite journal
author = Shoemake, K.
year = 1985
title = Animating rotation with quaternion curves
journal = Proceedings of the 12th annual conference on Computer graphics and interactive techniques
pages = 245–254
doi = 10.1145/325334.325242
url = http://portal.acm.org/citation.cfm?id=325242] , in which heused the concept of a quaternion Cite book
author = Bottema, O.; Roth, B.
year = 1990
url = http://books.google.co.uk/books?id=f8I4yGVi9ocC
publisher = Dover Publications
isbn =0486663469
tile= Theoretical kinematics] for rotation interpolation. A detailed list of references on this topic can be found in cite journal
author = Roschel, O.
year = 1998
title = Rational motion design—a survey
journal = Computer-Aided Design
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doi = 10.1016/S0010-4485(97)00056-0
url = http://www.ingentaconnect.com/content/els/00104485/1998/00000030/00000003/art00056] and cite journal
author = Purwar, A.; Ge, Q. J.
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title = On the effect of dual weights in computer aided design of rational motions
journal = ASME Journal of Mechanical Design
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url = http://scitation.aip.org/getabs/servlet/GetabsServlet?prog=normal&id=JMDEDB000127000005000967000001&idtype=cvips&gifs=yes88] .

Rational Bezier and B-Spline Motions

Let $hat\; \{\; extbf\{q\; =\; extbf\{q\}\; +\; varepsilon\; extbf\{q\}^0$denote a unit dual quaternion. A homogeneous dual quaternion may bewritten as a pair of quaternions, $hat\; \{\; extbf\{Q=\; extbf\{Q\}\; +varepsilon\; extbf\{Q\}^0$; where $extbf\{Q\}\; =\; w\; extbf\{q\},\; extbf\{Q\}^0\; =\; w\; extbf\{q\}^0\; +\; w^0\; extbf\{q\}$. This is obtained byexpanding $hat\; \{\; extbf\{Q\; =\; hat\; \{w\}\; hat\; \{\; extbf\{q$ using
dual number algebra (here, $hat\{w\}=w+varepsilon\; w^0$).

In terms of dual quaternions and the homogeneous coordinates of a point $extbf\{P\}:(P\_1,\; P\_2,\; P\_3,\; P\_4)$ of the object, the transformation equation in terms of quaternions is given by (see Who|date=July 2008] for details)

$ilde\; \{\; extbf\{P\; =\; extbf\{Q\}\; extbf\{P\}\; extbf\{Q\}^ast\; +\; P\_4\; [(\; extbf\{Q\}^0)\; extbf\{Q\}^ast\; -\; extbf\{Q\}(\; extbf\{Q\}^0)^ast]\; ,$where $extbf\{Q\}^ast$ and $(\; extbf\{Q\}^0)^ast$ areconjugates of $extbf\{Q\}$ and $extbf\{Q\}^0$, respectively and$ilde\; \{\; extbf\{P$ denotes homogeneous coordinates of the pointafter the displacement.

Given a set of unit dual quaternions and dual weights $hat\{\; extbf\{q\_i,\; hat\; \{w\}\_i;\; i\; =\; 0...n$ respectively, thefollowing represents a rational Bezier curve in the space ofdual quaternions.

$hat\{\; extbf\{Q(t)\; =\; sumlimits\_\{i\; =\; 0\}^n\; \{B\_i^n\; (t)hat\; \{\; extbf\{Q\_i\}\; =sumlimits\_\{i\; =\; 0\}^n\; \{B\_i^n\; (t)hat\; \{w\}\_i\; hat\{\; extbf\{q\_i\}$

where $B\_i^n(t)$ are the Bernstein polynomials. The Bezier dual quaternion curve given by above equation defines a rational Bezier motion ofdegree $2n$.

Similarly, a B-spline dual quaternion curve, which defines a NURBSmotion of degree $2p$, is given by,

$hat\; \{\; extbf\{Q(t)\; =sumlimits\_\{i\; =\; 0\}^n\; \{N\_\{i,p\}(t)\; hat\; \{\; extbf\{Q\_i\; \}\; =sumlimits\_\{i\; =\; 0\}^n\; \{N\_\{i,p\}(t)\; hat\; \{w\}\_i\; hat\; \{\; extbf\{q\_i\; \}$

where $N\_\{i,p\}(t)$ are the$p$th-degree B-spline basis functions.

A representation for the rational Bezier motion and rationalB-spline motion in the Cartesian space can be obtained bysubstituting either of the above two preceding expressions for $hat\; \{\; extbf\{Q(t)$ in the equation for point transform. In what follows, we deal with the case of rational Bezier motion. The, the trajectory of a point undergoing rational Beziermotion is given by,

$ilde\; \{\; extbf\{P^\{2n\}(t)\; =\; [H^\{2n\}(t)]\; extbf\{P\},$

$H^\{2n\}(t)]\; =\; sumlimits\_\{k\; =\; 0\}^\{2n\}\{B\_k^\{2n\}(t)\; [H\_k]\; \},$

where $[H^\{2n\}(t)]$ is the matrixrepresentation of the rational Bezier motion of degree$2n$ in Cartesian space. The following matrices$[H\_k\; ]$ (also referred to as Bezier ControlMatrices) define the "affine control structure" of the motion:

$[H\_k]\; =\; frac\{1\}\{C\_k^\{2nsumlimits\_\{i+j=k\}\{C\_i^n\; C\_j^n\; w\_i\; w\_j\; [H\_\{ij\}^ast]\; \},$

where $[H\_\{ij\}^ast]\; =\; [H\_i^+]\; [H\_j^-]\; +\; [H\_j^-]\; [H\_i^\{0+\}]\; -\; [H\_i^+]\; [H\_j^\{0-\}\; ]\; +\; (alpha\_i\; -\; alpha\_j)\; [H\_j^-]\; [Q\_i^+]$.

In the above equations, $C\_i^n$ and $C\_j^n$are binomial coefficients and $alpha\_i\; =\; w\_i^0/w\_i,\; alpha\_j\; =w\_j^0/w\_j$ are the weight ratios and

In above matrices, $(q\_\{i,1\},\; q\_\{i,2\},\; q\_\{i,3\},\; q\_\{i,4\})$are four components of the real part $(\; extbf\{q\}\_i)$ and$(q\_\{i,1\}^0,\; q\_\{i,2\}^0,\; q\_\{i,3\}^0,\; q\_\{i,4\}^0)$ are fourcomponents of the dual part$(\; extbf\{q\}\_i^0)$ of the unitdual quaternion $(hat\; \{\; extbf\{q\_i)$.

Example

References

External links

* [http://cadcam.eng.sunysb.edu/ Computational Design Kinematics Lab]
* [http://my.fit.edu/~pierrel/ Robotics and Spatial Systems Laboratory (RASSL)]
* [http://synthetica.eng.uci.edu:16080/~mccarthy/ Robotics and Automation Laboratory]

ee also

* Quaternion and Dual quaternion
* NURBS
* Computer Animation
* Robotics
* Robot kinematics
* Computational Geometry
* CNC machining
* Mechanism design