Fixed Point Iteration-based Problem Solution without the Calculation of the Jacobian

Hamza Khan, J. Tar

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Abstract

In technical applications typical problems arise that can be "differentially" treated by the calculation and subsequent inversion of certain Jacobi matrices. However, in many cases, the calculation of the inverse means no necessary computations because the solution is interesting only for a given, well defined input array, while by the use of the (generalized) inverse it can be obtained for an arbitrary input. In this view a matrix inversion-free quasi-differential solution of the inverse kinematic task of robots was recently suggested that showed nice and stable behavior in and in the vicinity of the kinematic singularities where the traditional, matrix-inversion-based approaches have difficulties and need complementary tricks to remain stable. In the case of a redundant robot this approach requires the calculation of the Jacobian. In the present paper it is shown that in the special case of quadratic Jacobians even the calculation of the Jacobian can be omitted. The present approach may have significance in the development of novel Receding Horizon Controllers in which replacement of Lagrange's original Reduced Gradient Method was suggested with a simple Fixed Point Iteration for the gradient of the auxiliary function of the optimization under constraints, because in it only quadratic Jacobians occur. As a simple example a 2 degree of freedom arm is considered via simulations to reveal the behavior of this approach. It is shown that by the use of a simple complementary norm reduction built in into the solution this approach is promising.

Original languageEnglish
Title of host publicationSAMI 2019 - IEEE 17th World Symposium on Applied Machine Intelligence and Informatics, Proceedings
PublisherInstitute of Electrical and Electronics Engineers Inc.
Pages187-192
Number of pages6
ISBN (Electronic)9781728102504
DOIs
Publication statusPublished - Jan 1 2019
Event17th IEEE World Symposium on Applied Machine Intelligence and Informatics, SAMI 2019 - Herl'any, Slovakia
Duration: Jan 24 2019Jan 26 2019

Publication series

NameSAMI 2019 - IEEE 17th World Symposium on Applied Machine Intelligence and Informatics, Proceedings

Conference

Conference17th IEEE World Symposium on Applied Machine Intelligence and Informatics, SAMI 2019
CountrySlovakia
CityHerl'any
Period1/24/191/26/19

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Keywords

  • Fixed Point Iterations
  • Inverse Kinematic Task
  • Open Kinematic Chain
  • Receding Horizon Controllers

ASJC Scopus subject areas

  • Artificial Intelligence
  • Computer Science Applications
  • Information Systems
  • Control and Optimization
  • Human-Computer Interaction

Cite this

Khan, H., & Tar, J. (2019). Fixed Point Iteration-based Problem Solution without the Calculation of the Jacobian. In SAMI 2019 - IEEE 17th World Symposium on Applied Machine Intelligence and Informatics, Proceedings (pp. 187-192). [8782749] (SAMI 2019 - IEEE 17th World Symposium on Applied Machine Intelligence and Informatics, Proceedings). Institute of Electrical and Electronics Engineers Inc.. https://doi.org/10.1109/SAMI.2019.8782749