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-spline motions. Ge and Ravani,^[2][3] 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,^[4] in which he used the concept of a quaternion^[5] for rotation interpolation. A detailed list of references on this topic can be found in ^[6] and.^[7]

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

In terms of dual quaternions and the homogeneous coordinates of a point ${\displaystyle {\textbf {P}}:(P_{1},P_{2},P_{3},P_{4})}$  of the object, the transformation equation in terms of quaternions is given by

${\displaystyle {\tilde {\textbf {P}}}={\textbf {Q}}{\textbf {P}}{\textbf {Q}}^{\ast }+P_{4}[({\textbf {Q}}^{0}){\textbf {Q}}^{\ast }-{\textbf {Q}}({\textbf {Q}}^{0})^{\ast }],}$  where ${\displaystyle {\textbf {Q}}^{\ast }}$  and ${\displaystyle ({\textbf {Q}}^{0})^{\ast }}$  are conjugates of ${\displaystyle {\textbf {Q}}}$  and ${\displaystyle {\textbf {Q}}^{0}}$ , respectively and ${\displaystyle {\tilde {\textbf {P}}}}$  denotes homogeneous coordinates of the point after the displacement.^[7]

Given a set of unit dual quaternions and dual weights ${\displaystyle {\hat {\textbf {q}}}_{i},{\hat {w}}_{i};i=0...n}$  respectively, the following represents a rational Bézier curve in the space of dual quaternions.

${\displaystyle {\hat {\textbf {Q}}}(t)=\sum \limits _{i=0}^{n}{B_{i}^{n}(t){\hat {\textbf {Q}}}_{i}}=\sum \limits _{i=0}^{n}{B_{i}^{n}(t){\hat {w}}_{i}{\hat {\textbf {q}}}_{i}}}$ 

where ${\displaystyle B_{i}^{n}(t)}$  are the Bernstein polynomials. The Bézier dual quaternion curve given by above equation defines a rational Bézier motion of degree ${\displaystyle 2n}$ .

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

${\displaystyle {\hat {\textbf {Q}}}(t)=\sum \limits _{i=0}^{n}{N_{i,p}(t){\hat {\textbf {Q}}}_{i}}=\sum \limits _{i=0}^{n}{N_{i,p}(t){\hat {w}}_{i}{\hat {\textbf {q}}}_{i}}}$ 

where ${\displaystyle N_{i,p}(t)}$  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 ${\displaystyle {\hat {\textbf {Q}}}(t)}$  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,

${\displaystyle {\tilde {\textbf {P}}}^{2n}(t)=[H^{2n}(t)]{\textbf {P}},}$ 

${\displaystyle H^{2n}(t)]=\sum \limits _{k=0}^{2n}{B_{k}^{2n}(t)[H_{k}]},}$ 

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

${\displaystyle [H_{k}]={\frac {1}{C_{k}^{2n}}}\sum \limits _{i+j=k}{C_{i}^{n}C_{j}^{n}w_{i}w_{j}[H_{ij}^{\ast }]},}$ 

where ${\displaystyle [H_{ij}^{\ast }]=[H_{i}^{+}][H_{j}^{-}]+[H_{j}^{-}][H_{i}^{0+}]-[H_{i}^{+}][H_{j}^{0-}]+(\alpha _{i}-\alpha _{j})[H_{j}^{-}][Q_{i}^{+}]}$ .

In the above equations, ${\displaystyle C_{i}^{n}}$  and ${\displaystyle C_{j}^{n}}$  are binomial coefficients and ${\displaystyle \alpha _{i}=w_{i}^{0}/w_{i},\alpha _{j}=w_{j}^{0}/w_{j}}$  are the weight ratios and

${\displaystyle [H_{j}^{-}]=\left[{\begin{array}{rrrr}q_{j,4}&-q_{j,3}&q_{j,2}&-q_{j,1}\\q_{j,3}&q_{j,4}&-q_{j,1}&-q_{j,2}\\-q_{j,2}&q_{j,1}&q_{j,4}&-q_{j,3}\\q_{j,1}&q_{j,2}&q_{j,3}&q_{j,4}\\\end{array}}\right],}$ 

${\displaystyle [Q_{i}^{+}]=\left[{\begin{array}{rrrr}0&0&0&q_{i,1}\\0&0&0&q_{i,2}\\0&0&0&q_{i,3}\\0&0&0&q_{i,4}\\\end{array}}\right],}$ 

${\displaystyle [H_{i}^{0+}]=\left[{\begin{array}{rrrr}0&0&0&q_{i,1}^{0}\\0&0&0&q_{i,2}^{0}\\0&0&0&q_{i,3}^{0}\\0&0&0&q_{i,4}^{0}\\\end{array}}\right],}$ 

${\displaystyle [H_{j}^{0-}]=\left[{\begin{array}{rrrr}0&0&0&-q_{j,1}^{0}\\0&0&0&-q_{j,2}^{0}\\0&0&0&-q_{j,3}^{0}\\0&0&0&q_{j,4}^{0}\\\end{array}}\right],}$ 

${\displaystyle [H_{i}^{+}]=\left[{\begin{array}{rrrr}q_{i,4}&-q_{i,3}&q_{i,2}&q_{i,1}\\q_{i,3}&q_{i,4}&-q_{i,1}&q_{i,2}\\-q_{i,2}&q_{i,1}&q_{i,4}&q_{i,3}\\-q_{i,1}&-q_{i,2}&-q_{i,3}&q_{i,4}\\\end{array}}\right].}$ 

In above matrices, ${\displaystyle (q_{i,1},q_{i,2},q_{i,3},q_{i,4})}$  are four components of the real part ${\displaystyle ({\textbf {q}}_{i})}$  and ${\displaystyle (q_{i,1}^{0},q_{i,2}^{0},q_{i,3}^{0},q_{i,4}^{0})}$  are four components of the dual part${\displaystyle ({\textbf {q}}_{i}^{0})}$  of the unit dual quaternion ${\displaystyle ({\hat {\textbf {q}}}_{i})}$ .

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

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