Conceptual design of Schoenflies motion generators based on the wrench graph

The subject of this paper is about the conceptual design of parallel Schoenflies motion generators based on the wrench graph. By using screw theory and Grassmann geometry, some conditions on both the constraint and the actuation wrench systems are generated for the assembly of limbs of parallel Schoenflies motion generators, i.e., 3T1R parallel manipulators. Those conditions are somehow related to the kinematic singularities of the manipulators. Indeed, the parallel manipulator should not be in a constraint singularity in the starting configuration for a valid architecture, otherwise it cannot perform the required motion pattern. After satisfying the latter condition, the parallel manipulator should not be in an actuation singularity in a general configuration, otherwise the obtained parallel manipulator is permanently singular. Based on the assembly conditions, six types of wrench graphs are identified and correspond to six typical classes of 3T1R parallel manipulators. The geometric properties of these six classes are highlighted. A simplified expression of the superbracket decomposition is obtained for each class, which allows the determination and the comparison of the singularities of 3T1R parallel manipulators at their conceptual design stage. The methodology also provides new architectures of parallel Schoenflies motion generators based on the classification of wrench graphs and on their singularity conditions. INTRODUCTION The primary concern of the conceptual design is the generation of physical solutions to meet certain design specification [1]. The concept generated at this phase affects the basic shape generation and material selection of the product concerned. In the subsequent phase of detailed design, it becomes exceedingly difficult, or even impossible to make a correction the shortcomings of a poor design concept formulated at the conceptual design stage [2]. Therefore, the conceptual design of parallel manipulators (PMs) is crucial task which aims at defining the architectures of the associated kinematic chains. In this paper, the focus of conceptual design process is the Schoenflies Motion Generators (SMGs). The parallel manipulators are called Schoenflies Motion Generators if they can perform four degree of freedom (dof ) displacements of a rigid body. These motions involve three independent translations and one rotation about a fixed axis [3]. This set of displacements was first studied by the German mathematicianmineralogist Arthur Moritz Schoenflies (1853-1928). Over the past few decades, the creation of various designs of 3T1R parallel manipulators were broaden, especially after huge success of the Quattro [4]. Gogu discovered an isotropic architecture (its jacobian matrix is diagonal and constant), named the Isoglide4 [5, 6], which is composed of four legs with prismatic actuators. Another topology within the same family was introduced by Gosselin [7], named the Quadrupteron. The symmetrical design was proposed by Angeles [3], namely the McGill SMG, with two identical legs which in turn decreases the number of joints. There exist two architecture varieties of H4 family, either with revolute or prismatic actuators [8–10]. The H4 robot designed by Pierrot et al. [8, 9], is a fully-parallel mechanism with no passive kinematic chain between the base and the nacelle. This idea brought out the mechanism with four legs. Each revolute joint in the leg is actuated. Whereas the H4 robot with prismatic actuators mounted to the base, was presented by Wu et al. [10]. Another mechanism constructed by three identical legs was proposed by Briot and Bonev [11, 12], called Pantopteron-4, where each leg comprises a pantograph linkage. Since it only employs three legs, the Pantopteron-4 gains great advantage in terms of workspace volume and acceleration capacities. The type synthesis approach based upon screw theory is widely used for generating many parallel manipulators as shown by Kong and Gosselin [13]. This approach allows us to produce numerous kinematic chains, by discovering the wrench system W that is reciprocal to the twist system T of the moving platform. Based upon the reciprocity condition, Joshi and Tsai developed a procedure to express Jacobian matrix J of limited dof parallel manipulators, comprises both constraint and actuation wrenches [14]. In this paper, this matrix is named the extended Jacobian matrix JE . The rows of JE are composed of n linearly independent actuation wrenches plus (6-n) linearly independent constraint wrenches. These wrenches correspond to six Plücker lines, composing J. The determinant of this J is equal to the superjoin of 6 Plücker lines, named superbracket of Grassmann-Cayley Algebra. It allows a translation of synthetic geometric conditions into invariant (coordinate-free) algebraic expression [15]. The superbracket decomposition was employed by BenHorin and Shoham [16–18] to analyse the singularity of 6 dof 1 ha l-0 08 30 40 0, v er si on 1 4 Ju n 20 13 Author manuscript, published in "ASME 2013 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, United States (2013)"