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A univariate marginal distribution algorithm based on extreme elitism and its application to the robotic inverse displacement problem

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Abstract

In this paper, a univariate marginal distribution algorithm in continuous domain (UMDA C ) based on extreme elitism (EEUMDA C ) is proposed for solving the inverse displacement problem (IDP) of robotic manipulators. This algorithm highlights the effect of a few top best solutions to form a primary evolution direction and obtains a fast convergence rate. Then it is implemented to determine the IDP of a 4-degree-of-freedom (DOF) Barrett WAM robotic arm. After that, the algorithm is combined with differential evolution (EEUMDA C -DE) to solve the IDP of a 7-DOF Barrett WAM robotic arm. In addition, three other heuristic optimization algorithms (enhanced leader particle swarm optimization, intersect mutation differential evolution, and evolution strategies) are applied to find the IDP solution of the 7-DOF arm and their performance is compared with that of EEUMDA C -DE.

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Acknowledgment

This work has been supported by research grants from the Natural Sciences and Engineering Research Council (NSERC) of Canada, the Canada Foundation for Innovation (CFI), the British Columbia Knowledge Development Fund (BCKDF) and the Canada Research Chair in Mechatronics and Industrial Automation held by Clarence W. de Silva.

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Authors and Affiliations

  1. Industrial Automation Laboratory, Department of Mechanical Engineering, University of British Columbia, 6250 Applied Science Lane, Vancouver, BC, V6T 1Z4, Canada

    Shujun Gao & Clarence W. de Silva

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  1. Shujun Gao
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  2. Clarence W. de Silva
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Correspondence to Shujun Gao.

Appendix

Appendix

The appendix shows the expression of \( \Delta \) p and \( \Delta o \) of a numerical example on the 7-DOF Barrett WAM arm. Here c i  = cos(\( \theta \) i ), s i  = sin(\( \theta \) i ), i = 1, 2,…,7.

$$ \begin{aligned} \Delta p(\theta_{1} ,\theta_{2} , \ldots ,\theta_{7} ) = (0.55c_{1} s_{2} - 0.045s_{1} s_{3} + 0.045c_{3} (s_{1} s_{3} - c_{1} c_{2} c_{3} ){\kern 1pt} - 0.3s_{4} (s_{1} s_{3} {\kern 1pt} - c_{1} c_{2} c_{3} ) - {\kern 1pt} 0.06s_{6} (c_{5} (c_{4} (s_{1} s_{3} - c_{1} c_{2} c_{3} ) + c_{1} s_{2} s_{4} ) + {\kern 1pt} s_{5} (c_{3} \hfill \\ s_{1} + c_{1} c_{2} s_{3} )) - 0.06c_{6} (s_{4} (s_{1} s_{3} - c_{1} c_{2} c_{3} ) - c_{1} c_{4} s_{2} ) + 0.3c_{1} c_{4} s_{2} + 0.045c_{1} s_{2} s_{3} {\kern 1pt} + 0.045c_{1} c_{2} c_{{3{\kern 1pt} }} - 0.2401)^{2} {\kern 1pt} + (0.045c_{1} s_{3} + 0.55s_{1} s_{2} {\kern 1pt} + 0.06 \hfill \\ s_{{6{\kern 1pt} }} (c_{{5{\kern 1pt} }} (c_{4} (c_{1} s_{3} + {\kern 1pt} c_{2} c_{3} s_{1} ) - s_{1} s_{{2{\kern 1pt} }} s_{4} ) + s_{5} (c_{1} c_{{3{\kern 1pt} }} - c_{{2{\kern 1pt} }} s_{{1{\kern 1pt} }} s_{3} )){\kern 1pt} + 0.06c_{6} (s_{{4{\kern 1pt} }} (c_{{1{\kern 1pt} }} s_{3} + c_{{2{\kern 1pt} }} c_{3} s_{1} ){\kern 1pt} + c_{4} s_{1} s_{2} ){\kern 1pt} - 0.045c_{3} (c_{1} s_{{3{\kern 1pt} }} + c_{2} c_{3} s_{1} ) + 0.3s_{4} (c_{1} s_{3} + c_{2} c_{3} \hfill \\ s_{1} ) + 0.045c_{2} c_{3} s_{1} + 0.3c_{4} s_{1} s_{2} + 0.045s_{1} s_{2} s_{3} - 0.2486)^{2} {\kern 1pt} + (0.55c_{2} + 0.3c_{2} c_{4} + 0.045c_{2} s_{3} - 0.045c_{3} s_{2} - 0.06s_{6} (c_{5} (c_{2} s_{4} + c_{3} c_{4} s_{2} ) - s_{2} {\kern 1pt} \hfill \\ s_{3} s_{5} ) + 0.06c_{6} (c_{2} c_{4} - c_{3} {\kern 1pt} s_{{2{\kern 1pt} }} s_{4} ){\kern 1pt} + 0.045c_{3}^{2} s2 - 0.3c_{3} s_{2} s_{4} - 0.8382)^{2} \hfill \\ \end{aligned} $$
$$\begin{aligned} & \Delta o(\theta_{1} ,\theta_{2} , \ldots ,\theta_{7} ) = (0.55c_{1} s_{2} - 0.045s_{1} s_{3} + 0.045c_{3} (s_{1} s_{3} - c_{1} c_{2} c_{3} ) - 0.3s_{4} (s_{1} s_{3} - c_{1} c_{2} c_{3} ) - 0.06s_{6} (c_{5} (c_{4} (s_{1} s_{3} - c_{1} c_{2}c_{3} ) + c_{1} s_{2} s_{4} ) + s_{5} (c_{3} s_{1} \\ & \quad +c_{1} c_{2} s_{3} ))- 0.06c_{6} (s_{4} (s_{1} s_{3} - c_{1} c_{2}c_{3} )- c_{1} c_{4} s_{2} )+ 0.3c_{1} c_{4} s_{2} + 0.045c_{1}s_{2} s_{3} + 0.045c_{1} c_{2} c_{3} - 0.2401)^{2} + (0.7788c_{7}(s_{5} (c_{4} (c_{1} s_{3} + c_{2} c_{3} s_{1} )- s_{1} s_{2} s_{4}) \\&\quad - c_{5} (c_{1} c_{3} - c_{2} s_{1} s_{3} )) + 0.4040c_{7}(s_{5} (c_{4} (s_{1} s_{3} - c_{1} c_{2} c_{3} ) - c_{1} s_{2} s_{4})- c_{5} (c_{3} s_{1} + c_{1} c_{2} s_{3} )) + 0.4799s_{7} (c_{6}(c_{5} (c_{2} s_{4} + c_{3} c_{4} s_{2} ) - s_{2} s_{3} s_{5} )+s_{6} (c_{2} c_{4} - c_{3} s_{2} \\&\quad s_{4} )) + 0.4799 c_{7}(s_{5} (c_{2} s_{4} + c_{3} c_{4} s_{2} ) + c_{{5}} s_{2} s_{3} )+0.404s_{7} (c_{6} (c_{5} (c_{4} (s_{1} s_{3} - c_{1} c_{2} c_{3} ) +c_{1} s_{2} s_{4} ) + s_{5} (c_{3} s_{1} + c_{1} c_{2} s_{3} )) -s_{6} (s_{4} (s_{1} s_{3} - c_{1} c_{2} c_{3} ) - c_{1} c_{4}s_{2} )) \\&\quad + 0.7788s_{7} (c_{6} (c_{5} (c_{4} (c_{1} s_{3}+ c2c_{3} s_{1} ) - s_{1} s_{2} s_{4} ) + s_{5} (c_{1} c_{3} -c_{2} s_{1} s_{3} )) - s_{6} (s_{4} (c_{1} s_{3} + c_{2} c_{3}s_{1} ) + c_{4} s_{1} s_{2} )) + 1.0)^{2} + (0.4150s_{6} (c_{5}(c_{4} (c_{1} s_{3} + c_{2} c_{3} s_{1} ) \\&\quad - s_{1} s_{2}s_{4} ) + s_{5} (c_{1} c_{3} - c_{2} s_{1} s_{3} )) - 0.8771s_{6}(c_{5} (c_{2} s_{4} + c_{3} c_{4} s_{2} ) - s_{2} s_{3} s_{5} ) +0.4150 c_{6} (s_{4} (c_{{1 }} s_{3} + c_{2} c_{3} s_{1} ) +c_{4} s_{1} s_{2} ) + 0.8771c_{6} (c_{2} c_{4} - c_{3} s_{2}s_{4} ) + \\&\quad 0.2419 s_{6} (c_{5} (c_{4} (s_{1} s_{3} -c_{1} c_{2} c_{3} ) + c_{1} s_{2} s_{4} ) + s_{5} (c_{3} s_{1} +c_{1} c_{2} s_{3} )) + 0.2419c_{6} (s_{4} (s_{1} s_{3} - c_{1}c_{2} c_{3} ) - c_{1} c_{4} s_{2} ) - 1.0)^{2} + (0.4704s_{7}(s_{5} (c_{4} (c_{1} s_{3} + c_{2} c_{3} \\&\quad s_{1} ) - s_{1}s_{2} s_{4} ) - c_{5} (c_{1} c_{3} - c_{2} s_{1} s_{3} )) -0.8822s_{7} (s_{5} (c_{4} (s_{1} s_{3} - c_{1} c_{2} c_{3} ) +c_{1} s_{2} s_{4} ) - c_{5} (c_{3} s_{1} + c_{{1 }} c_{2} s_{3})) + 0.0207c_{7} (c_{6} (c_{5} (c_{2} s_{4} + c_{3} c_{4} s_{2} ) -s_{2} s_{3} s_{5} ) + s_{6} \\&\quad (c_{2} c_{4} - c_{3} s_{2}s_{4} )) + 0.8822c_{7} (c_{6} (c_{5} (c_{4} (s_{{1 }} s_{3} -c_{1} c_{2} c_{3} ) + c_{1} s_{2} s_{4} ) + s_{5} (c_{3} s_{1} +c_{1} c_{2} s_{3} )) - s_{6} (s_{4} (s_{1} s_{3} - c_{1} c_{2}c_{3} ) - c_{1} c_{4} s_{2} )) - 0.0207s_{7} (s_{5} (c_{2} s_{4}+ c_{3} c_{4} s_{2} ) + \\&\quad c_{5} s_{2} s_{2} s_{3} ) -0.4704c_{7} (c_{6} (c_{5} (c_{4} (c_{1} c_{1} s_{3} + c_{2} c_{3}s_{1} ) - s_{1} s_{2} s_{4} ) + s_{5} (c_{1} c_{3} - c_{2} s_{1}s_{3} )) - s_{6} (s_{4} (c_{1} s_{3} + c_{2} c_{{2 }} c_{3} s_{1} )+ c_{4} s_{1} s_{2} )) + 1.0)^{2} + (0.045c_{1} s_{3} + 0.55\\&\quad s_{1} s_{2} + 0.06s_{6} (c_{5} (c_{4} (c_{1} s_{3} +c_{2} c_{3} s_{1} ) - s_{1} s_{2} s_{4} ) + s_{5} (c_{1} c_{3} -c_{2} c_{2} s_{1} s_{3} )) + c_{6} (s_{4} (c_{1} s_{3} + c_{{2 }}c_{3} s_{1} ) + c_{4} s_{1} s_{2} ) - 0.045 c_{3} (c_{1} c_{1}s_{3} + c_{2} c_{2} c_{3} s_{1} ) + 0.3 s_{4} (c_{{1 }} s_{3} +c_{2} \\&\quad c_{3} c_{3} s_{1} ) + 0.045c_{2} c_{3} s_{1} +0.3c_{4} s_{1} s_{2} + 0.045 s_{1} s_{2} s_{3} - 0.2486)^{2} +(0.55c_{2} + 0.3c_{2} c_{4} + 0.045c_{2} s_{3} - 0.045 c_{3}s_{2} - 0.06s_{6} (c_{5} (c_{2} s_{4} + c_{3} c_{4} s_{2} ) -s_{2} s_{3} s_{5} ) \\&\quad + 0.06c6(c_{2} c_{4} - c_{3}s_{2} s_{4} ) + 0.045c_{3}^{2} s_{2} - 0.3c_{3} s_{2} s_{4} -0.8382)^{2} \end{aligned}$$

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Gao, S., de Silva, C.W. A univariate marginal distribution algorithm based on extreme elitism and its application to the robotic inverse displacement problem. Genet Program Evolvable Mach 18, 283–312 (2017). https://doi.org/10.1007/s10710-017-9298-8

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