- Rational motion
In

kinematics , the motion of arigid 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 byrational function s (ratio of twopolynomial function s) of time. They produce rationaltrajectories , and therefore they integrate well with the existingNURBS (Non-Uniform Rational B-Spline) based industry standardCAD/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 ofcurves andsurfaces , methods have been developed forcomputer aided design of rational motions.These CAD methods for motion design find applications in

animation in computer graphics (key frameinterpolation ), trajectory planning inrobotics (taught-position interpolation), spatial navigation invirtual reality , computer aided geometric design of motion via interactive interpolation,CNC tool path planning , and task specification inmechanism 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] ofspatial 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 aquaternion 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] forrotation 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

volume = 30

issue = 3

pages = 169–178

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.

year = 2005

title = On the effect of dual weights in computer aided design of rational motions

journal = ASME Journal of Mechanical Design

volume = 127

issue = 5

pages = 967-972

doi = 10.1115/1.1906263

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

$[H\_j^-]\; =\; left\; [\; egin\{array\}\{rrrr\}q\_\{j,4\}\; -q\_\{j,3\}\; q\_\{j,2\}\; -q\_\{j,1\}\; \backslash q\_\{j,3\}\; q\_\{j,4\}\; -q\_\{j,1\}\; -q\_\{j,2\}\; \backslash -q\_\{j,2\}\; q\_\{j,1\}\; q\_\{j,4\}\; -q\_\{j,3\}\; \backslash q\_\{j,1\}\; q\_\{j,2\}\; q\_\{j,3\}\; q\_\{j,4\}\; \backslash end\{array\}\; ight]\; ,$

$[Q\_i^+]\; =\; left\; [\; egin\{array\}\{rrrr\}0\; 0\; 0\; q\_\{i,1\}\; \backslash 0\; 0\; 0\; q\_\{i,2\}\; \backslash 0\; 0\; 0\; q\_\{i,3\}\; \backslash 0\; 0\; 0\; q\_\{i,4\}\; \backslash end\{array\}\; ight]\; ,$

$[H\_i^\{0+\}]\; =\; left\; [egin\{array\}\{rrrr\}0\; 0\; 0\; q\_\{i,1\}^0\; \backslash 0\; 0\; 0\; q\_\{i,2\}^0\; \backslash 0\; 0\; 0\; q\_\{i,3\}^0\; \backslash 0\; 0\; 0\; q\_\{i,4\}^0\; \backslash end\{array\}\; ight]\; ,$

$[H\_j^\{0-\}]\; =\; left\; [egin\{array\}\{rrrr\}0\; 0\; 0\; -q\_\{j,1\}^0\; \backslash 0\; 0\; 0\; -q\_\{j,2\}^0\; \backslash 0\; 0\; 0\; -q\_\{j,3\}^0\; \backslash 0\; 0\; 0\; q\_\{j,4\}^0\; \backslash end\{array\}\; ight]\; ,$

$[H\_i^+]\; =\; left\; [\; egin\{array\}\{rrrr\}q\_\{i,4\}\; -q\_\{i,3\}\; q\_\{i,2\}\; q\_\{i,1\}\; \backslash q\_\{i,3\}\; q\_\{i,4\}\; -q\_\{i,1\}\; q\_\{i,2\}\; \backslash -q\_\{i,2\}\; q\_\{i,1\}\; q\_\{i,4\}\; q\_\{i,3\}\; \backslash -q\_\{i,1\}\; -q\_\{i,2\}\; -q\_\{i,3\}\; q\_\{i,4\}\; \backslash end\{array\}\; ight]\; .$

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 andDual quaternion

*NURBS

*Computer Animation

*Robotics

*Robot kinematics

*Computational Geometry

*CNC machining

*Mechanism design

*Wikimedia Foundation.
2010.*

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