Rational motion

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 q ^ = q + ε q 0 denote a unit dual quaternion. A homogeneous dual quaternion may be written as a pair of quaternions, Q ^ = Q + ε Q 0 ; where Q = w q , Q 0 = w q 0 + w 0 q . This is obtained by expanding Q ^ = w ^ q ^ using dual number algebra (here, w ^ = w + ε w 0 ).

In terms of dual quaternions and the homogeneous coordinates of a point P : ( P 1 , P 2 , P 3 , P 4 ) of the object, the transformation equation in terms of quaternions is given by (see for details)

P ~ = Q P Q ∗ + P 4 [ ( Q 0 ) Q ∗ − Q ( Q 0 ) ∗ ] , where Q ∗ and ( Q 0 ) ∗ are conjugates of Q and Q 0 , respectively and P ~ denotes homogeneous coordinates of the point after the displacement.

Given a set of unit dual quaternions and dual weights q ^ i , w ^ i ; i = 0... n respectively, the following represents a rational Bézier curve in the space of dual quaternions.

Q ^ ( t ) = ∑ i = 0 n B i n ( t ) Q ^ i = ∑ i = 0 n B i n ( t ) w ^ i q ^ i

where B i n ( t ) are the Bernstein polynomials. The Bézier dual quaternion curve given by above equation defines a rational Bézier motion of degree 2 n .

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

where N i , p ( t ) 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 Q ^ ( t ) in the equation for point transform. In what follows, we deal with the case of rational Bézier motion. The, the trajectory of a point undergoing rational Bézier motion is given by,

where [ H 2 n ( t ) ] is the matrix representation of the rational Bézier motion of degree 2 n in Cartesian space. The following matrices [ H k ] (also referred to as Bézier Control Matrices) define the affine control structure of the motion:

where [ H i j ∗ ] = [ H i + ] [ H j − ] + [ H j − ] [ H i 0 + ] − [ H i + ] [ H j 0 − ] + ( α i − α j ) [ H j − ] [ Q i + ] .

In the above equations, C i n and C j n are binomial coefficients and α i = w i 0 / w i , α 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 ( q i ) and ( q i , 1 0 , q i , 2 0 , q i , 3 0 , q i , 4 0 ) are four components of the dual part ( q i 0 ) of the unit dual quaternion ( q ^ i ) .