Latest Advances in Robot Kinematics
This book is of interest to researchers inquiring about modern topics and methods in the kinematics, control and design of robotic manipulators. It considers the full range of robotic systems, including serial, parallel and cable driven manipulators
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Abstract The determination of rigid or overconstrained subsystems is an important task in the creative design of robotic mechanisms and in the processing of CADgenerated models. While for planar mechanisms with planar graphs a number of methods have been proposed, the case of general spatial mechanisms is still an open topic. In this paper, a novel method for identifying rigid subsystems is presented. The method uses the independent loops as building blocks of a graph, called kinematical network, which describes the overall transmission behavior. The detection of rigid subsystems can then be realized by finding the minimal cutsets in the solution flow of the kinematical network. The method is independent of the subspace in which the bodies are moving, i.e., it is possible to mix planar, spherical and spatial systems. Moreover, it is fast, as only the implicitly coupled loops need to be processed, which comprise much less elements than the number of bodies. Key words: Rigid subsystems, degrees of freedom, degenerate kinematic chains
1 Introduction The topic of rigid subsystem detection has attracted scientists in the robotics and mechanisms community for many decades. The problem is to detect whether the overall Gr¨ubler sum of degrees of freedom (DoF) for the mechanism is composed by substructures with positive DoF (i.e. movable subsystems) and negative DoF (i.e. overconstrained subsystems). Such cases arise for example in the automatic generation of candidate mechanisms in creative design of robotic devices [3], or when the constraint graphs for the mechanisms are automatically generated by CAD systems [10]. An example is shown in Fig. 1. Here, a loop with an internal DoF = 1 is attached to a subsystem of internal DoF = −1. Thus, the overall Gr¨ubler count suggests that the mechanism has DoF = 0, which is not correct, as the system Shuxian Xia · Huafeng Ding · Andres Kecskemethy Chair for Mechanics and Robotics, University of Duisburg-Essen, Duisburg, Germany, e-mail: [email protected], [email protected], [email protected] J. Lenarˇciˇc, M. Husty (eds.), Latest Advances in Robot Kinematics, DOI 10.1007/978-94-007-4620-6 3, © Springer Science+Business Media Dordrecht 2012
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Fig. 1 Example of a rigid subchain.
actually has a DoF of 1 and the lower subsystem can be exchanged by a single rigid body. Starting with the rigidity test by Laman’s theorem [9], a number of methods for planar mechanisms [2, 5, 6, 12, 15], as well as simple spatial systems [10, 11] have been proposed. However, as shown in [14], these methods are restricted to mechanisms with planar graphs, and they also do not cover the case of general spatial mechanisms with arbitrary joints. Ding et al. [4] presented an algorithm for detecting rigid subchains for planar mechanisms with also non-planar graphs. They use the “smallest” independent loop as a starting point of a set of stepwise extended clusters for which rigidity is tested. As a rigid subsystem does not necessarily contain the smallest loop, the algorithm o
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