Journal of Rehabilitation Robotics, 2018, 6, 8-21 8 E-ISSN: 2308-8354/18 © 2018 Synergy Publishers Dual Quaternions Robotics: A) The 3R Planar Manipulator Mahmoud Gouasmi * , Belkacem Gouasmi and Mohammed Ben-Ahmed-Dahou Algeria Structural Mechanics Research Laboratory, Mechanical Engineering Department, Blida University, Algeria Abstract: Kinematics analysis studies the relative motions, such as, first of all, the displacement in space of the end effector of a given robot, and thus its velocity and acceleration, associated with the links of the given robot that is usually designed so that it can position its end-effector with a three degree-of-freedom of translation and three degree-of-freedom of orientation within its workspace. This chapter presents mainly, on the light of both main concepts; the first being the screw motion or/ and dual quaternions kinematics while the second concerns the classical ‘Denavit and Hartenberg parameters method’ the direct kinematics of a planar manipulator. First of all, examples of basic solid movements such as rotations, translations, their combinations and general screw motions are studied using both (4x4) matrices rigid body transformations and dual quaternions so that the reader could compare and note the similarity of the results obtained using one or the other method. Both dual quaternions technique as well as its counterpart the classical ‘Denavit and Hartenberg parameters method’ are finally applied to a three degree of freedom (RRR) planar manipulator. Finally, we and the reader, can observe that the two methods confirm exactly one another by giving us the same results for each of the examples and applications considered, while noting that the fastest, simplest more straightforward and easiest to apply method, is undoubtedly the one using dual quaternions. As a result this work may as well act as a beginners guide to the practicality of using dual-quaternions to represent the rotations and translations ie: or any rigid motion in character-based hierarchies. We must emphasize the fact that the use of Matlab software and quaternions and / or dual quaternions in the processing of 3D rotations and/or screw movements is and will always be the most efficient, fast and accurate first choice. Dual quaternion direct kinematics method could be generalised, in the future, to more complicated spatial and/ or industrial robots as well as to articulated and multibody systems. Keyword: Dual Quaternions, Forward Kinematics, Homogeneous Matrix, Screw Motion. 1. INTRODUCTION Many research students have a great deal of trouble understanding essentially what quaternions are [1-3] and how they can represent rotation. So when the subject of dual-quaternions is presented, it is usually not welcomed with open arms. Dual-quaternions are a break from the norm (i.e., matrices) which we hope to entice the reader into supporting willingly to represent their rigid transforms. The reader should walk away from this chapter with a clear understanding of what dual-quaternions are and how they can be used [4]. First we begin with a short recent and related work that emphasises the power of dual-quaternions: The dual-quaternion has been around since 1882 [5-7] but has gained less attention compared to quaternions alone ; while the most recent work which has taken hold and has demonstrated the practicality of dual-quaternions, both in robotics and computer graphics can be resumed in: - Kavan [8] demonstrated the advantages of dual-quaternions in character skinning and blending. - Ivo [9] extended Kavan’s work with dual-quaternions and q-tangents as an alternative method for representing rigid transforms instead of matrices, and gives evidence that the results can be faster with accumulated transformations of joints if the inferences per vertex are large enough. - Selig [10] addressed the key problem in computer games. - Vasilakis [11] discussed skeleton-based rigid-skinning for character animation. - Kuang [12] presented a *Address correspondence to this author at the Algeria Structural Mechanics Research Laboratory, Mechanical Engineering Department, Blida University, Algeria; E-mail: ygouasmi@hotmail.com strategy for creating real-time animation of clothed body movement. -Pham [13] solved linked chain inverse kinematic (IK) problems using Jacobian matrix in the dual-quaternion space. -Malte [14] used a mean of multiple computational (MMC) model with dual-quaternions to model bodies. - Ge [15] demonstrated dual-quaternions to be an efficient and practical method for interpolating three-dimensional motions. -Yang -Hsing [16] calculated the relative orientation using dual-quaternions. - Perez [17] formulated dynamic constraints for articulated robotic systems using dual-quaternions. - Further reading on the subject of dual numbers and derivatives is presented by Gino [18]. In the last three decades, the field of robotics has widened its range of applications, due to recent developments in the major domains of robotics like kinematics, dynamics and control, which leads to the sudden growth of robotic applications in areas such as manufacturing, medical surgeries, defense, space vehicles, under-water explorations etc. To use robotic manipulators in real-life applications, the first step is to obtain the accurate kinematic model [19]. In this context, a lot of research has been carried out in the literature, which leads to the evolution of new modeling schemes along with the refinement of existing methodologies describing the kinematics of robotic manipulators. Screw theory based solution methods have been widely used in many robotic applications .The elements of screw theory can be traced to the work of Chasles and Poinsot [20, 21], in the early 1800’s and Whittaker [22]. Using the theorems of Chasles and Poinsot as a