A Survey of Mathematical Tools in Topology and Performance Integrated Modeling and Design of Robotic Mechanism

Abstract

Topology and performance are the two main topics dealt in the development of robotic mechanisms. However, it is still a challenge to connect them by integrating the modeling and design process of both parts in a unified frame. As the properties associated with topology and performance, finite motion and instantaneous motion of the robot play key roles in the procedure. On the purpose of providing a fundamental preparation for integrated modeling and design, this paper carries out a review on the existing unified mathematic frameworks for motion description and computation, involving matrix Lie group and Lie algebra, dual quaternion and pure dual quaternion, finite screw and instantaneous screw. Besides the application in robotics, the review of the work from these mathematicians concentrates on the description, composition and intersection operations of the finite and instantaneous motions, especially on the exponential-differential maps which connect the two sides. Furthermore, an in-depth discussion is worked out by investigating the algebraical relationship among these methods and their further progress in integrated robotic development. The presented review offers insightful investigation to the motion description and computation, and therefore would help designers to choose appropriate mathematical tool in the integrated design and modeling and design of mechanisms and robots.

An Approach to Determine All Joint Motions of Parallel Robot and Its Software Development

“…, ,1 , 2 ,, 5 ff i i   SS (28) Since the RSR limbs are less than 6-DoF, the equations are overdetermined. The initial values are obtained by genetic algorithm firstly.…”

“…All the variables of both actuated and passive joints would be solved in the meanwhile. However, the redundant parameters in the motion description may complicate the computation process [28]. To the best of our knowledge, there is no available software for the inverse kinematics of parallel robots based on the existing methods now.…”

“…In our previous work, finite screw theory has been applied for motion description and calculation [29][30][31][32]. Compared with Lie groups and dual quaternions, the algebraic structure of finite screw, having only 6 elements, is more compact, and the combination or intersection operation is more concise [28]. Motivated by the inherent merits, a concise and general method for the inverse kinematics of parallel robots would be proposed based on finite screw.…”

Mathematical Modeling Methods and Their Application in the Analysis of Complex Signal Systems

“…In a narrow sense, a mathematical model refers specifically to a structure of mathematical relations that reflects only a specific system or a specific problem. The definition of a mathematical model given by the well-known Chinese mathematician Jiang Qiyuan is that it is a mathematical structure that makes the necessary simplifications for a specific object or purpose in the real world according to its intrinsic laws and later expresses it by applying appropriate mathematical tools [4][5][6][7]. From the above descriptions, it can be seen that different mathematicians have defined mathematical models in terms of their types, modeling processes, and modeling methods.…”