William Kingdon Clifford (1845–1879)

Abstract

Abstract. William Kingdon Clifford was an English mathematician and philosopher who worked extensively in many branches of pure mathematics and classical mechanics. Although he died young, he left a deep and long-lasting legacy, particularly in geometry. One of the main achievements that he is remembered for is his pioneering work on integrating Hamilton's Elements of Quaternions with Grassmann's Theory of Extension into a more general coherent corpus, now referred to eponymously as Clifford algebras. These geometric algebras are utilised in engineering mechanics (especially in robotics) as well as in mathematical physics (especially in quantum mechanics) for representing spatial relationships, motions, and dynamics within systems of particles and rigid bodies. Clifford's study of geometric algebras in both Euclidean and non-Euclidean spaces led to his invention of the biquaternion, now used as an efficient representation for twists and wrenches in the same context as that of Ball's Theory of Screws. Biographical NotesWilliam Kingdon Clifford was a 19th Century English mathematician and scientific philosopher who, though he lived a short life, produced major contributions in many areas of mathematics, mechanics, physics and philosophy. This he achieved during a mere fifteen-year professional career. He was the archetypal polymath, since as well as displaying remarkable mathematical skills, he was also an accomplished literature and classics scholar. Clifford was fluent in reading French, German and Spanish, as he considered these to be important for his mathematical work. He learned Greek, Arabic and Sanskrit for the challenge they presented, Egyptian hieroglyphics as an intellectual exercise, and Morse code and shorthand, because he wished to understand as many forms for communicating ideas as possible. During his lifetime Clifford was energetic and influential in championing the scientific method in

Forward Kinematics Algorithm in Dual Quaternion Space Based on Denavit-Hartenberg Convention

“…Dual numbers were introduced by an English polymath William Kingdon Clifford in the nineteenth century [26]. A dual number is considered a generalized type of complex number where the imaginary number i is replaced with the operator ε. Dual numbers are defined as: 𝑎 = 𝑝 + 𝑑𝜀, (5) where ε ≠ 0 and ε 2 = 0.…”

“…( 5) are considered primary and dual parts of the dual number a, respectively. Dual numbers are essentially an ordered pair of primary and dual parts (p,d) with distinct arithmetic operations given in Table 2 [26][27].…”

Rigid–Flexible Coupled System Attitude–Orbit Integration Fixed-Time Control

“…Compared with other methods that can represent spiral motion in space, dual quaternions have a concise and compact form with high computational efficiency. Roonye's research has shown that dual quaternions are the most effective and concise form of spatial line transformation [22,23].…”

Half-turns and line symmetric motions

“…This work seems to have appeared first in Clifford's 1871 paper, "Preliminary sketch of biquaternions" [5] but see also [20] for more details on the history. In Clifford's paper several different 'biquaternions' are considered, these are characterised by the properties of the extra generator (ω), introduced.…”