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.
Highlights
IntroductionMechanism, serving as the execution unit, is one of the essential subsystems of robot
The development of robot meeting the requirements from application scenarios depends largely on the analysis and design of robotic mechanism, which focus on topology and performance [1, 2]
There are three mathematical tools that have been applied to the descriptions, computations and mappings of finite and instantaneous motions, i.e. matrix Lie group and Lie algebra [16], dual quaternion and pure dual quaternion [17], finite screw and instantaneous screw [18]
Summary
Mechanism, serving as the execution unit, is one of the essential subsystems of robot. There are three mathematical tools that have been applied to the descriptions, computations and mappings of finite and instantaneous motions, i.e. matrix Lie group and Lie algebra [16], dual quaternion and pure dual quaternion [17], finite screw and instantaneous screw [18]. It has been rigorously proved that a differential map exists in the finite and instantaneous screws These three mathematical tools have been applied at different stages of mechanism development, their capabilities in unifying the topology and performance analysis and design have not been realized.
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