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Enhancing the performance in the offline controller tuning of robotic manipulators with chaos: a comparative study with differential evolution

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Abstract

The decision-making process regarding utilizing specific chaotic maps as number generators and the different ways they can be incorporated into differential evolution (DE), as well as the type of system and task for which chaos is functional in the controller tuning problem, is an open issue. This paper aims to address this need for more knowledge in the field by proposing sixteen different chaotic DE variants and presenting a formal study on the tuning of PID controllers through these variants for regulation and tracking tasks with robotic systems of incremental complexity. These systems include planar open-chain robotic manipulators with one, two, and six revolute joints. Concerning the proposed variants, they utilize pseudo-random numbers obtained from well-known chaotic maps, including Logistic, Sine, Hénon, and Lozi, at various DE stages such as initialization, mutation, crossover, and in all those stages. The comparative results based on descriptive and nonparametric statistics with respect to the proposed chaotic variants of DE and the DE using a general-purpose pseudo-random number generator such as Mersenne Twister (MT) indicates that using the single-state chaotic maps, especially the Sine map, increases the performance of the search process when used in the crossover stage or throughout the algorithm in complex systems. It is also observed that Sine map-based chaotic numbers can be useful enough during the search initialization for problems involving relatively less complex systems. Additionally, considering that chaotic maps require simple arithmetic operations, it is deduced that the variants that provide better results than the widely used MT also avoid the intricate operations involved in such a generator, reducing the computational burden to a good extent and increasing the area of application where this is a limitation.

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Data are available on request from the corresponding author.

Abbreviations

\(\alpha \) :

Control parameter for the Hénon and Lozi maps

\(\bar{\varvec{q}}\) :

Vector of desired angular positions

\({\bar{q}}_i\) :

Desired angular position for the ith link

\(\beta \) :

Control parameter for the Hénon and Lozi maps

\(\varvec{e}\) :

Vector of angular position errors

\(\varvec{F}\) :

Non-conservative forces vector

\(\varvec{g}\) :

Gravity vector

\(\varvec{q}\) :

Vector of angular positions

\(\varvec{u}\) :

Control input vector

\(\varvec{u}_{i}\) :

The ith offspring in differential evolution

\(\varvec{v}_{i}\) :

The ith mutant in differential evolution

\(\varvec{x}\) :

Vector of design variables

\(\varvec{x}_{\textrm{lb}}\) :

Lower-bound vector for the design variables

\(\varvec{x}_{ri}\) :

The ith individual from the population used during the mutation in differential evolution

\(\varvec{x}_{\textrm{ub}}\) :

Upper-bound vector for the design variables

\(\varvec{z}\) :

State vector

\(\varvec{z}_0\) :

Initial state vector

\(\dot{\bar{\varvec{q}}}\) :

Vector of desired angular velocities

\(\dot{{\bar{q}}}_i\) :

Desired angular velocity for the ith link

\(\dot{\varvec{e}}\) :

Vector of angular velocity errors

\(\dot{\varvec{q}}\) :

Vector of angular velocities

\(\dot{\varvec{z}}\) :

State equation

\({\dot{q}}_i\) :

Angular velocity of the ith link

\(\mu \) :

Control parameter for the Sine map

\(a\) :

Statistical significance for the Wilcoxon test

\(C\) :

Coriolis and centrifugal forces matrix

\(CR\) :

Crossover probability for differential evolution

\(dt\) :

Sampling time interval

\(F\) :

Scaling factor for differential evolution

\(G\) :

Current generation in differential evolution

\(g\) :

Acceleration of gravity in the Earth

\(G_{\max }\) :

Maximum number of generations for differential evolution

\(H_0\) :

Null hypothesis for the Wilcoxon test

\(H_a\) :

Alternative hypothesis for the Wilcoxon test

\(I_i\) :

Inertia of the ith link

\(ISE\) :

Integral squared error

\(J\) :

Objective function

\(j_\textrm{rand}\) :

The jth design variable used in the binomial crossover of differential evolution

\(K_d\) :

Diagonal matrix of derivative gains

\(K_i\) :

Diagonal matrix of integral gains

\(K_p\) :

Diagonal matrix of proportional gains

\(k_{dj}\) :

Derivative gain to calculate the jth control input

\(k_{ij}\) :

Integral gain to calculate the jth control input

\(k_{pj}\) :

Proportional gain to calculate the jth control input

\(l_{ci}\) :

Length to the mass center of the ith link

\(l_{i}\) :

Length of the ith link

\(M\) :

Inertia matrix

\(m_i\) :

Mass of the ith link

\(n\) :

Degrees of freedom

\(NP\) :

Population size for differential evolution

\(q_i\) :

Angular position of the ith link

\(r\) :

Control parameter for the Logistic map

\(t\) :

Time

\(t_f\) :

Final simulation time

\(u_i\) :

The ith control input

\(X_G\) :

Current population in differential evolution

\(z_{i,n+1}\) :

Next value for the ith state of a chaotic map

\(z_{i,n}\) :

Current value for the ith state of a chaotic map

\(z_{i}\) :

The ith state of a chaotic map

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Acknowledgements

Authors acknowledge the support of the Consejo Nacional de Humanidades, Ciencias y Tecnologías de México (CONAHCyT), the Universidad Autónoma de la Ciudad de México (UACM), and the Secretaría de Investigación y Posgrado (SIP) of the Instituto Politécnico Nacional (IPN).

Funding

This research was funded in part by the Universidad Autónoma de la Ciudad deMéxico (UACM) and by the Secretaría de Investigación y Posgrado (SIP) of the Instituto Politécnico Nacional (IPN).

Author information

Authors and Affiliations

  1. Research and Postgraduate Division, Tecnológico Nacional de México/IT de Tlalnepantla, 54070, Estado de México, Mexico

    Miguel Fernando Parra-Ocampo

  2. Unidad Profesional Interdisciplinaria de Ingeniería Campus Guanajuato, Instituto Politécnico Nacional, 36275, Guanajuato, Mexico

    Omar Serrano-Pérez

  3. Colegio de Ciencia y Tecnología, Universidad Autónoma de la Ciudad de México, 06720, Mexico City, Mexico

    Alejandro Rodríguez-Molina, Gerardo Hernández & Máximo E. Sánchez-Gutiérrez

  4. Centro de Innovación y Desarrollo Tecnológico en Cómputo, Instituto Politécnico Nacional, 07700, Mexico City, Mexico

    Miguel Gabriel Villarreal-Cervantes & Víctor Manuel Silva-García

Authors
  1. Miguel Fernando Parra-Ocampo
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  2. Omar Serrano-Pérez
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  3. Alejandro Rodríguez-Molina
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Contributions

M.F.P.-O., O.S.-P., and A.R.-M. contributed to conceptualization; M.F.P.-O., A.R.- M., and M.G.V.-C contributed to methodology; G.H., M.E.S.-G., and V.M.S.-G. contributed to software; A.R.-M., M.G.V.-C., and V.M.S.-G. performed validation; M.G.V.-C and V.M.S.-G. carried out formal analysis; M.F.P.-O., O.S.-P., and M.G.V.-C performed investigation; O.S.-P. and M.G.V.-C contributed to resources; G.H. and M.E.S.-G. performed data curation; M.F.P.-O., A.R.-M., and M.G.V.-C. performed writing—original draft preparation; A.R.-M., G.H., and M.E.S.-G. performed writing—review and editing; G.H., M.E.S.-G., and V.M.S.-G. contributed to visualization; A.R.-M. and M.G.V.-C. performed supervision; A.R.-M. and M.G.V.-C. performed project administration; O.S.-P. and M.G.V.-C contributed to funding acquisition. All authors have read and agreed to the submitted version of the manuscript.

Corresponding authors

Correspondence to Alejandro Rodríguez-Molina or Miguel Gabriel Villarreal-Cervantes.

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The authors declare no conflict of interest.

Supplementary Information

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Supplementary file 1 (pdf 250 KB)

Appendix A: Detailed results of the Wilcoxon signed-rank pairwise tests

Appendix A: Detailed results of the Wilcoxon signed-rank pairwise tests

Tables 8, 9, and 10 present the results of the Wilcoxon test for the R, 2R, and 6R manipulators, respectively. The tables include the task carried out by the manipulators, the DE variant, the Wilcoxon test, and the outcome of the test conducted on the distributions of J obtained at 0%, 25%, 50%, 75%, and 100% of the total available \(G_{\max }\) generations in DE. The results feature the p values of each test, representing the probability of accepting \(H_0\) and rejecting \(H_a\). Thus, when a p value \(<a\), \(H_a\) is accepted. The tables also display the sum of ranks calculated in each test. Each p value is accompanied by \((+)\) when the first alternative in the test outperforms the second, with \((-)\) when the opposite is true, and with \((\approx )\) when the two alternatives show no significant difference.

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Parra-Ocampo, M.F., Serrano-Pérez, O., Rodríguez-Molina, A. et al. Enhancing the performance in the offline controller tuning of robotic manipulators with chaos: a comparative study with differential evolution. Int. J. Dynam. Control (2024). https://doi.org/10.1007/s40435-024-01423-6

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