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It is well known that the automorphism group Aut(H) of the algebra of real quaternions H consists entirely of inner automorphisms iq : p → q ·p·q-1 for invertible q ∈ H and is isomorphic to the group of rotations SO(3). Hence, H has only inner derivations D = ad(x), x ∈ H. See [4] for derivations of various types of quaternions over the reals. Unlike real quaternions, the algebra Hd of dual quaternions has no nontrivial inner derivation. Inspired from almost inner derivations for Lie algebras, which were first introduced in [3] in their study of spectral geometry, we introduce coset invariant derivations for dual quaternion algebra being a derivation that simply keeps every dual quaternion in its coset space. We begin with finding conditions for a linear map on Hd become a derivation and show that the dual quaternion algebra Hd consists of only central derivations. We also show how a coset invariant central derivation of Hd is closely related with its spectrum. [ABSTRACT FROM AUTHOR]

We present a new formulation method to solve the kinematic problem of multiarm robot systems. Our major aims were to formulize the kinematic problem in a compact closed form and avoid singularity problems in the inverse kinematic solution. The new formulation method is based on screw theory and quaternion algebra. Screw theory is an effective way to establish a global description of a rigid body and avoids singularities due to the use of the local coordinates. The dual quaternion, the most compact and efficient dual operator to express screw displacement, was used as a screw motion operator to obtain the formulation in a compact closed form. Inverse kinematic solutions were obtained using Paden-Kahan subproblems. This new formulation method was implemented into the cooperative working of 2 Stäubli RX160 industrial robot-arm manipulators. Simulation and experimental results were derived. [ABSTRACT FROM AUTHOR]