Robotica (2016) volume 34, pp. 942–956. © Cambridge University Press 2014 doi:10.1017/S0263574714001982 A new approach to singularity-free inverse kinematics using dual-quaternionic error chains in the Davies method Andre Schneider de Oliveira† ∗ , Edson Roberto De Pieri‡, Ubirajara Franco Moreno‡ and Daniel Martins§ †Department of Informatics, Federal University of Technology – Parana, Curitiba, PR, Brazil ‡Department of Automation and Systems, Federal University of Santa Catarina, Florianopolis, SC, Brazil E-mails: edson@das.ufsc.br, moreno@das.ufsc.br §Department of Mechanical Engineering, Federal University of Santa Catarina, Florianopolis, SC, Brazil E-mail: daniel@emc.ufsc.br (Accepted June 20, 2014. First published online: July 24, 2014) SUMMARY The manipulation in singular regions promotes an instantaneous reduction in mechanism mobility, which can result in some disturbances in the trajectory tracking. The application of the quaternionic elements for motion representation not only guarantees an orthonormal transformation but also results in the smallest variance and minimizes the acceleration peaks. The use of a unit quaternion avoids these phenomena, but there are dimensional limitations that make it impossible to translate the representation. This work presents a methodology for using dual quaternions in the analysis of robot kinematics using the Davies method, which avoids kinematic singularities and ensures the optimal torque profiles. KEYWORDS: Mechanism mobility; Kinematic singularities; Davies method; Dual quaternions; Optimal torque profiles. 1. Introduction In robotics, the kinematics perform the conversion between the joint and operational spaces. Some restrictions, which are directly related to the mechanism structure, define a nonlinear mapping between these spaces. A temporary phenomenon (i.e., a limited period of time) exists that is directly related to mechanism pose and introduces nonlinearities in this transformation; this phenomenon is called the singularity. Singularities represent configurations where the structure mobility is reduced, i.e., as in ref. [1], and it is not possible to impose an arbitrary motion to an end effector. The singularities can be divided into two types: boundary singularities (also called as geometric singularities), which occur near the extension or contraction limits, and internal singularities (also called kinematic singularities), which occur within the reachable space of the manipulator and are usually caused by the alignment of two or more axes or by particular configurations of the end effector. The consequence of this effect is that in an internal singularity, there are infinite solutions for the inverse kinematics. In proximity to the singularities, small velocities of the end effectors generate high speeds in the joints due to the gradual reduction of the mobility. The occurrence of singularities frequently happens on mobile base systems where it is possible to accomplish the reorientation of the whole structure. In manipulator-vehicle systems where a * Corresponding author. E-mail: andreoliveira@utfpr.edu.br