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The design of a robotic manipulator begins with the dimensioning of its various links to meet performance specifications. However, a methodology for the determination of the manipulator architecture, i.e., the fundamental geometry of the links, regardless of their shapes, is still lacking. Attempts have been made to apply the classical paradigms of linkage synthesis for motion generation, as in the Burmester Theory. The problem with this approach is that it relies on a specific task, described in the form of a discrete set of end-effector poses, which kills the very purpose of using robots, namely, their adaptability to a family of tasks. Another approach relies on the minimization of a condition number of the Jacobian matrix over the architectural parameters and the posture variables of the manipulator. This approach is not trouble-free either, for the matrices involved can have entries that bear different units, the matrix singular values thus being of disparate dimensions, which prevents the evaluation of any version of the condition number. As a means to cope with dimensional inhomogeneity, the concept of characteristic length was put forth. However, this concept has been slow in finding acceptance within the robotics community, probably because it lacks a direct geometric interpretation. In this paper the concept is revisited and put forward from a different point of view. In this vein, the concept of homogeneous space is introduced in order to relieve the designer from the concept of characteristic length. Within this space the link lengths are obtained as ratios, their optimum values as well as those of all angles involved being obtained by minimizing a condition number of the dimensionally homogeneous Jacobian. Further, a comparison between the condition number based on the two-norm and that based on the Frobenius norm is provided, where it is shown that the use of the Frobenius norm is more suitable for design purposes. Formulation of the inverse problem—obtaining link lengths—and the direct problem—obtaining the characteristic length of a given manipulator—are described. Finally a geometric interpretation of the characteristic length is provided. The application of the concept to the design and kinetostatic performance evaluation of serial robots is illustrated with examples.

In this paper, a novel parallel robot is introduced. The robot, a Schönflies-Motion Generator (SMG), is capable of a special class of motions, namely, those produced with serial robots termed SCARA, an acronym for Selective-Compliance Assembly Robot Arm. These motions involve three independent translations and one rotation about an axis of fixed direction. Such motions are known to form a subgroup of the displacement group of rigid-body motions, termed the Schönflies subgroup. The SMG is composed of two identical four-degree-of-freedom serial chains in a parallel array, sharing one common base and one common moving platform. The proximal module of each chain is active and has two controlled axes, the motors being installed on the fixed base. The links can thus be made light, thereby allowing for higher operational speeds. The distal module, in turn, is passive and follows the motions of its active counterpart, the whole mechanism giving, as a result, a four-degree-of-freedom motion to its end-platform.Keywords: Parallel Robot, Manipulator, Kinematics, SCARA, Schönflies displacement subgroup Historical Note and State of the ArtMost industrial robots have articulated structures of the serial type, where each link is coupled to two other links, except for the end links, which are coupled to one single neighbour. However, these simple, open kinematic chains of multi-axis machines exhibit some drawbacks, mostly due to the pyramidal effect: each axis must carry all other axes located upstream in the chain. For this reason, parallel robots have attracted the attention of researchers and end-users. First prototypes of the parallel machines date back to the middle of the 20th century. Gough's machine was built in the early 50s and fully operational in 1954. Stewart produced a paper in 1965 (Stewart, 1965); however, he had never built any real prototype (Bonev, 2003). The history of the development of modern robots with a parallel structure can be traced back to the early eighties (Merlet, 2006). As a milestone in this regard, Karl-Erik Neumann designed and built a new type of machine, the "parallel kinematic robot", in 1987robot", in (Brumson, 2005. In spite of the early invention of parallel robots, they were not used widely until the early 90's, since their architectures demanded sophisticated control not affordable with the computational 1 means available at the time. After 1992, when Comau Pico launched the first multiprocessor controller, new control systems appeared in the market, capable to run parallel-kinematics machines (PKM). The six-rod machining centers (TM series) of Lapik (Russia) appeared on the market in 1994, and measuring machines of similar kinematics (KIM series) in 1992 (the development of KIMs started in the early 80s) (Lapic, 1994).Most PKM are based on hexapods, also known as Gough-Stewart platforms (Stewart, 1965). An example of PKM currently used in industry is displayed in Fig. 1.The most widespread operation of PKMs is machining. This includes milling, cutting, drilling and deburri...

Abstract. In this paper, we introduce a new class of degenerate Hermite poly-Bernoulli polynomials and give some identities of these polynomials related to the Stirling numbers of the second kind. Some implicit summation formulae and general symmetry identities are derived by using different analytical means and applying generating functions. These results extend some known summations and identities of degenerate Hermite poly-Bernoulli numbers and polynomials.Mathematics subject classification (2010): 11B68, 11B73, 11B83, 33C45.

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