Extended Jacobian Inverse Kinematics and Approximation of Distributions

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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A Limited Interval Gradient Projection Algorithm for Inverse Kinematics

Advanced Materials Research ◽

10.4028/www.scientific.net/amr.490-495.7 ◽

2012 ◽

Vol 490-495 ◽

pp. 7-12

Author(s):

Wei Song ◽

Guang Hu

Keyword(s):

Inverse Kinematics ◽

Nonlinear Function ◽

Gradient Projection ◽

Projection Algorithm ◽

Motion Generation ◽

Local Linearization ◽

Gradient Projection Algorithm ◽

Inverse Kinematics Problem ◽

Jacobian Inverse ◽

Programming Time

The Jacobian inverse(JI) method is a well-known algorithms used for inverse kinematics solutions in motion generation. JI algorithm can be easily implemented, but it can generate singularity problems and it is not straight forward to implement constraints in the JI method. This paper presents a novel gradient projection algorithm that can convert the inverse kinematics problem to a constraint nonlinear programming problem. Meanwhile, by changing the programming time of each frame, local linearization of the nonlinear function and limited interval computing can be achieved simultaneously. Experimental results are presented to show the performance benefits of the proposed algorithm over JI methods.

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Design of inverse kinematics algorithms: extended Jacobian approximation of the dynamically consistent Jacobian inverse

Archives of Control Sciences ◽

10.1515/acsc-2015-0003 ◽

2015 ◽

Vol 25 (1) ◽

pp. 35-50 ◽

Cited By ~ 5

Author(s):

Joanna Ratajczak

Keyword(s):

Inverse Kinematics ◽

Ritz Method ◽

Approximation Error ◽

Approximation Problem ◽

Cholesky Decomposition ◽

Redundant Manipulators ◽

Jacobian Inverse

Abstract The paper presents the approximation problem of the inverse kinematics algorithms for the redundant manipulators. We introduce the approximation of the dynamically consistent Jacobian by the extended Jacobian. In order to do that, we formulate the approximation problem and suitably defined approximation error. By the minimization of this error over a certain region we can design an extended Jacobian inverse which will be close to the dynamically consistent Jacobian inverse. To solve the approximation problem we use the Cholesky decomposition and the Ritz method. The computational example illustrates the theory.

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