Sizing a Serial Chain to Fit a Task Trajectory Using Clifford Algebra Exponentials

In this paper we formulate the “generalized inverse kinematics problem” for a spatial serial chain, where the goal is to determine values for structural parameters as well as for the joint parameters. The kinematics equations of the chain are formulated first using matrix exponentials and then cast into a form based on exponentials in a Clifford algebra. These equations contain the coordinates of the joint axes explicitly and have a systematic structure that can be exploited in their solution. As an example we fit a seven degree-of-freedom CCS chain to a 12 position task trajectory. In this problem, we can also specify desired values for the first two joint angles, and compute the structural parameters and the remaining joint angles.

[1]  Guilin Yang,et al.  Task-based optimization of modular robot configurations: minimized degree-of-freedom approach , 2000 .

[2]  Lung-Wen Tsai,et al.  Design of Dyads with helical, cylindrical, spherical, revolute and prismatic joints , 1972 .

[3]  Glen Mullineux,et al.  Modeling spatial displacements using Clifford algebra , 2004 .

[4]  Damien Chablat,et al.  An Interval Analysis Based Study for the Design and the Comparison of Three-Degrees-of-Freedom Parallel Kinematic Machines , 2004, Int. J. Robotics Res..

[5]  Chung Ha Suh.  On the Duality in the Existence of R-R Links for Three Positions , 1969 .

[6]  Bernard Roth,et al.  Design Equations for the Finitely and Infinitesimally Separated Position Synthesis of Binary Links and Combined Link Chains , 1969 .

[7]  Sridhar Kota,et al.  Generalized kinematic modeling of Reconfigurable Machine Tools , 2002 .

[8]  Clément Gosselin,et al.  Conceptual Design and Dimensional Synthesis of a Novel 2-DOF Translational Parallel Robot for Pick-and-Place Operations , 2004 .

[9]  Layne T. Watson,et al.  Generalized Linear Product Homotopy Algorithms and the Computation of Reachable Surfaces , 2004, J. Comput. Inf. Sci. Eng..

[10]  C. H. Suh,et al.  Kinematics and mechanisms design , 1978 .

[11]  McCarthy,et al.  Geometric Design of Linkages , 2000 .

[12]  Philippe Wenger.  Some guidelines for the kinematic design of new manipulators , 2000 .

[13]  John J. Craig,et al.  Introduction to Robotics Mechanics and Control , 1986 .

[14]  Carl D. Crane,et al.  Kinematic Analysis of Robot Manipulators , 1998 .

[15]  K. C. Gupta.  Kinematic Analysis of Manipulators Using the Zero Reference Position Description , 1986 .

[16]  Phillip J. McKerrow,et al.  Introduction to robotics , 1991 .

[17]  Constantinos Mavroidis,et al.  Solving the Geometric Design Problem of Spatial 3R Robot Manipulators Using Polynomial Homotopy Continuation , 2002 .

[18]  J. Michael McCarthy,et al.  Introduction to theoretical kinematics , 1990 .

[19]  Constantinos Mavroidis,et al.  Geometric design of spatial PRR manipulators , 2004 .

[20]  Kostas Daniilidis,et al.  Hand-Eye Calibration Using Dual Quaternions , 1999, Int. J. Robotics Res..

[21]  Alba Perez,et al.  Geometric design of RRP, RPR and PRR serial chains , 2005 .

[22]  Richard M. Murray,et al.  A Mathematical Introduction to Robotic Manipulation , 1994 .

[23]  L. W. Tsai,et al.  Robot Analysis: The Mechanics of Serial and Parallel Ma-nipulators , 1999 .

[24]  J. Michael McCarthy,et al.  Dual quaternion synthesis of constrained robotic systems , 2003 .

[25]  J. M. Hervé.  The Lie group of rigid body displacements, a fundamental tool for mechanism design , 1999 .