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Research Papers

Performance-Oriented Design of Inverse Kinematics Algorithms: Extended Jacobian Approximation of the Jacobian Pseudo-Inverse

[+] Author and Article Information
Joanna Karpińska

Institute of Computer Engineering, Control and Robotics,  Wrocław University of Technology, ul. Janiszewskiego 11/17, 50-372 Wrocław, Polandjoanna.karpinska@pwr.wroc.pl

Krzysztof Tchoń

Institute of Computer Engineering, Control and Robotics,  Wrocław University of Technology, ul. Janiszewskiego 11/17, 50-372 Wrocław, Polandkrzysztof.tchon@pwr.wroc.pl

J. Mechanisms Robotics 4(2), 021008 (Apr 12, 2012) (8 pages) doi:10.1115/1.4006192 History: Received July 10, 2011; Revised January 17, 2012; Published April 10, 2012; Online April 12, 2012

For redundant robotic manipulators, we study the design problem of Jacobian inverse kinematics algorithms of desired performance. A specific instance of the problem is addressed, namely the optimal approximation of the Jacobian pseudo-inverse algorithm by the extended Jacobian algorithm. The approximation error functional is derived for the coordinate-free representation of the manipulator’s kinematics. A variational formulation of the problem is employed, and the approximation error is minimized by means of the Ritz method. The optimal extended Jacobian algorithm is designed for the 7 degrees of freedom (dof) POLYCRANK manipulator. It is concluded that the coordinate-free kinematics representation results in more accurate approximation than the coordinate expression of the kinematics.

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Copyright © 2012 by American Society of Mechanical Engineers
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Figures

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Figure 1

View of the POLYCRANK manipulator

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Figure 2

Coordinate-free representation: convergence of q1  − q4 for extended Jacobian (top) and for Jacobian pseudo-inverse (bottom) algorithm

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Figure 3

Coordinate-free representation: convergence of q5  − q7 for extended Jacobian (top) and for Jacobian pseudo-inverse (bottom) algorithm

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Figure 4

Coordinate-free representation: comparison of instantaneous joint velocities

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Figure 5

Coordinate representation: convergence of q1  − q4 for extended Jacobian (top) and for Jacobian pseudo-inverse (bottom) algorithm

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Figure 6

Coordinate representation: convergence of q5  − q7 for extended Jacobian (top) and for Jacobian pseudo-inverse (bottom) algorithm

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Figure 7

Coordinate representation: comparison of instantaneous joint velocities

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