Rational motion explained

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 (E³), 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 Bézier and rational B-spline based curve representation schemes can be combined with dual quaternion representation ^[1] of spatial displacements to obtain rational Bézier and B-splinemotions. Ge and Ravani,^{[2] [3]} 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,^[4] in which heused the concept of a quaternion^[5] for rotation interpolation. A detailed list of references on this topic can be found in ^[6] and.^[7]

Rational Bézier and B-spline motions

Let

denote a unit dual quaternion. A homogeneous dual quaternion may bewritten as a pair of quaternions, ; where . This is obtained byexpanding usingdual number algebra (here, ).

In terms of dual quaternions and the homogeneous coordinates of a point

of the object, the transformation equation in terms of quaternions is given by where and areconjugates of and , respectively and denotes homogeneous coordinates of the point after the displacement.^[7]

Given a set of unit dual quaternions and dual weights

respectively, thefollowing represents a rational Bézier curve in the space ofdual quaternions.

where

are the Bernstein polynomials. The Bézier dual quaternion curve given by above equation defines a rational Bézier motion ofdegree .

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

where

are the pth-degree B-spline basis functions.

A representation for the rational Bézier motion and rational B-spline motion in the Cartesian space can be obtained by substituting either of the above two preceding expressions for

in the equation for point transform. In what follows, we deal with the case of rational Bézier motion. The trajectory of a point undergoing rational Bézier motion is given by,

where

is the matrixrepresentation of the rational Bézier motion of degree in Cartesian space. The following matrices (also referred to as Bézier ControlMatrices) define the affine control structure of the motion:

where

.

In the above equations,

and are binomial coefficients and are the weight ratios and

In above matrices,

are four components of the real part and are fourcomponents of the dual part of the unitdual quaternion .

See also

• Quaternion and Dual quaternion
• NURBS
• Computer animation
• Robotics
• Robot kinematics
• Computational geometry
• CNC machining
• Mechanism design

External links

• Computational Design Kinematics Lab
• Robotics and Spatial Systems Laboratory (RASSL)
• Robotics and Automation Laboratory

Notes and References

1. Book: McCarthy, J. M. . An Introduction to Theoretical Kinematics . 1990 . MIT Press Cambridge, MA, USA . 978-0-262-13252-7 .
2. Ge, Q. J. . Ravani, B. . 1994 . Computer-Aided Geometric Design of Motion Interpolants . Journal of Mechanical Design . 116 . 3 . 756–762 . 10.1115/1.2919447 .
3. Ge, Q. J. . Ravani, B. . 1994 . Geometric Construction of Bézier Motions . Journal of Mechanical Design . 116 . 3 . 749–755 . 10.1115/1.2919446 .
4. Book: Shoemake, K. . Proceedings of the 12th annual conference on Computer graphics and interactive techniques - SIGGRAPH '85 . Animating rotation with quaternion curves . 1985 . 1985 . 19 . 3 . 245–254 . 10.1145/325334.325242 . 978-0897911665 . free .
5. Book: Bottema, O. . Roth, B. . 1990 . Theoretical kinematics . . 978-0-486-66346-3 . Theoretical kinematics.
6. Röschel, O. . 1998 . Rational motion design—a survey . Computer-Aided Design . 30 . 3 . 169–178 . 10.1016/S0010-4485(97)00056-0 .
7. Purwar, A. . Ge, Q. J. . 2005 . On the effect of dual weights in computer-aided design of rational motions . Journal of Mechanical Design . 127 . 5 . 967–972 . 10.1115/1.1906263 .