Abstract
Mixed-position kinematic synthesis is used to not only reach a certain number of precision positions, but also impose certain instantaneous motion conditions at those positions. In the traditional approach, one end-effector twist is defined at each precision position in order to achieve better guidance of the end-effector along a desired trajectory. For one-degree-of-freedom linkages, that suffices to fully specify the trajectory locally. However, for systems with a higher number of degrees of freedom, such as robotic systems, it is possible to specify a complete higher-dimensional subspace of potential twists at particular positions. In this work, we focus on the 3R serial chain. We study the three-dimensional subspaces of twists that can be defined and set the mixed-position equations to synthesize the chain. The number and type of twist systems that a chain can generate depend on the topology of the chain; we find that the spatial 3R chain can generate seven different fully defined twist systems. Finally, examples of synthesis with several fully defined and partially defined twist spaces are presented. We show that it is possible to synthesize 3R chains for feasible subspaces of different types. This allows a complete definition of potential motions at particular positions, which could be used for the design of precise interaction with contact surfaces.
Highlights
Twist Space Synthesis of 3R Chains.Mixed-position synthesis, in which a combination of finite and infinitesimal positions are defined for the dimensional synthesis, appears as an extension of Burmester’s theory for planar motion [1]
The methodology to prove the feasibility of the 3R chain to generate these twist systems, at any or some particular configuration, is presented in Section 4, and it is used in the subsection below, in particular the matrix formulation presented in Equations (18)–(22)
The synthesis of spatial 3R chains with the prescribed twist space at particular configurations is studied in this work, extending the current technique of defining a single velocity for guidance along a trajectory
Summary
First- and second-order derivatives can describe contact conditions in the synthesis problem [10], associating first- and second-order kinematics to curvatures and forces at the points of contact [11]. This identification has mainly been used for the design of robotic hands. We explore the relation between the twist subspaces that can be defined in combination with finite positions, and the topology and geometry of the kinematic chain that imposes the restrictions on the motion. While algebraic solutions for the finite-position synthesis of spatial chains tends to be a very complex problem, it is possible to state a simpler set of equations for the mixed-position synthesis. The resulting set of equations can be solved using the numerical Groebner basis and eigensystem methods
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