Rational motion

In kinematics, the motion of a rigid body is defined as a continuous set of displacements. One-parameter motions can be defined
as 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 Bézier and rational B-spline based curve representation schemes can be combined with dual quaternion representation of spatial displacements to obtain rational Bézier and B-spline
motions. Ge and Ravani, developed a new framework for geometric constructions
of spatial motions by combining the concepts from kinematics and CAGD. Their work was built upon the seminal paper of Shoemake, in which he
used the concept of a quaternion for rotation interpolation. A detailed list of references on this topic can be found in and.
Rational Bézier and B-spline motions.
Let formula_1
denote a unit dual quaternion. A homogeneous dual quaternion may be
written as a pair of quaternions, formula_2; where formula_3. This is obtained by
expanding formula_4 using
dual number algebra (here, formula_5).
In terms of dual quaternions and the homogeneous coordinates of a point formula_6 of the object, the transformation equation in terms of quaternions is given by
formula_7
where formula_8 and formula_9 are
conjugates of formula_10 and formula_11, respectively and
formula_12 denotes homogeneous coordinates of the point after the displacement.
Given a set of unit dual quaternions and dual weights formula_13 respectively, the
following represents a rational Bézier curve in the space of
dual quaternions.
formula_14
where formula_15 are the Bernstein polynomials. The Bézier dual quaternion curve given by above equation defines a rational Bézier motion of
degree formula_16.
Similarly, a B-spline dual quaternion curve, which defines a NURBS
motion of degree 2"p", is given by,
where formula_18 are the "p"th-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 formula_19 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 formula_22 is the matrix
representation of the rational Bézier motion of degree
formula_16 in Cartesian space. The following matrices
formula_24 (also referred to as Bézier Control
Matrices) define the "affine control structure" of the motion:
where formula_26.
In the above equations, formula_27 and formula_28
are binomial coefficients and formula_29 are the weight ratios and
In above matrices, formula_35
are four components of the real part formula_36 and
formula_37 are four
components of the dual partformula_38 of the unit
dual quaternion formula_39.