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The dual quaternion plays an advantageous role in the field of robotic kinematics and dynamics due to its compact expression. It is well known that quaternion can be easily implemented as algebraic operations on vectors through matrix operators. Similarly, the dual matrix operators can convert dual quaternion operations to matrix operations. However, it has not received adequate attention, and thus there are very few applications at present. With the dual matrix operators of dual quaternion, this paper re-verifies the equivalency between the conjugate formula of unit dual quaternion and dual Euler–Rodrigues formula. To combine this equivalence with the homomorphic mapping of Lie groups, a theoretical correlation of the current hand-eye calibration methods is established. Motivated by the attempt to establish a more complete scheme for hand-eye calibration, a new simultaneous method based on the conjugate formula of dual quaternion is proposed. The method verifies that the rotation problem can be extended to the rigid-body transformation problem via the dual algebra. Finally, the simulation and experimental validations for the hand–eye calibration method are given. • The conjugate formula of unit dual quaternion is re-verified via the dual operations. • A new simultaneous method is defined for hand-eye calibration. • The theoretical correlation with the existing methods is established. [ABSTRACT FROM AUTHOR]

• An algebraic synthesis method is proposed to form novel multi-loop linkages for rotation scaling. • Multi-loop planar, spherical, and spatial linkages can be constructed using the proposed method. • The generated linkages can exactly transmit angular motion with prescribed transmission ratios. • Exact angular motion transmission between parallel, intersecting, or skew axes is achieved. • Transmission characteristics of the multi-loop linkages are proved by kinematics analysis. This paper deals with the algebraic synthesis and input-output analysis of one degree-of-freedom (1-DOF) multi-loop planar, spherical, and spatial linkages for increasing or decreasing angular motion with a desired constant transmission ratio between two adjacent parallel, intersecting, and skew axes respectively. Based on the rotation scaling model and motion polynomials over dual quaternions, an algebraic synthesis method including three procedures is proposed to exactly construct novel multi-loop linkages owning arbitrarily prescribed constant transmission ratios and input/output axes. To illustrate this method, several 1-DOF multi-loop planar, spherical, and spatial linkages for rotation scaling are synthesized by designating various input and output axes as well as transmission ratios. Taking some multi-loop planar, spherical, and spatial linkages as examples, input-output analysis is carried out to verify their transmission characteristics. The results demonstrate that the generated 1-DOF multi-loop linkages can indeed transmit motion with prescribed constant transmission ratios and input/output axes. This work provides a framework for further investigation on mechanisms performing specified tasks for motion transmission. [ABSTRACT FROM AUTHOR]

Abstract We discuss existence of factorizations with linear factors for (left) polynomials over certain associative real involutive algebras, most notably over Clifford algebras. Because of their relevance to kinematics and mechanism science, we put particular emphasis on factorization results for quaternion, dual quaternion and split quaternion polynomials. A general algorithm ensures existence of a factorization for generic polynomials over division rings but we also consider factorizations for non-division rings. We explain the current state of the art, present some new results and provide examples and counter examples. [ABSTRACT FROM AUTHOR]

This paper exploits screw theory expressed via unit dual quaternion representation and its algebra to formulate both the forward (position+velocity) kinematics and pose control of an n -dof robot arm in an efficient way. Efficiency is in less computer memory usage, in fast computation of the equations, in singularity-free representation of task space, in robustness to numerical errors, and in compactness of the representations. The formulation is simple, intuitive and straightforward to implement. We validated this formulation experimentally on a 7 dof robot arm. [ABSTRACT FROM AUTHOR]

In this paper, a new forward kinematic algorithm based on dual quaternion for a six-degree-of-freedom (DOF) general Stewart Platform is proposed. The algorithm is established after taking into account the pose parameters of joints and the D–H parameters of each branch chain, which yields a precise mathematic model with a total of 174 geometric parameters. The validity of the algorithm is tested using an inverse kinematic algorithm of a general serial manipulator. The forward kinematic algorithms for the 6-6R and 6-2RP3R mechanisms can be expressed by a unified mathematic model. The proposed algorithm can solve the difficult problem of forward kinematics for the unsolved 6-6R parallel manipulator and can also be applied to error analysis, error synthesis, kinematic calibration, stiffness analysis and the accurate kinematic simulation by analysing the effect of geometric parameters on the platform. [ABSTRACT FROM AUTHOR]