An Accelerated Diagonally Structured CG Algorithm for Nonlinear Least Squares and Inverse Kinematics

Yunus, Rabiu Bashir; Ben Ghorbal, Anis; Zainuddin, Nooraini; Ibrahim, Sulaiman Mohammed

doi:10.3390/math13172766

Open AccessArticle

An Accelerated Diagonally Structured CG Algorithm for Nonlinear Least Squares and Inverse Kinematics

by

Rabiu Bashir Yunus

^1,2,*

,

Anis Ben Ghorbal

³

,

Nooraini Zainuddin

¹ and

Sulaiman Mohammed Ibrahim

^4,5

¹

Department of Applied Science, Faculty of Science, Management & Computing, Universiti Teknologi PETRONAS, Bandar Seri Iskandar 32610, Perak Darul Ridzuan, Malaysia

²

Department of Mathematics, Faculty of Computing and Mathematical Sciences, Aliko Dangote University of Science and Technology, Wudil 713101, Kano, Nigeria

³

Department of Mathematics and Statistics, Faculty of Science, Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh 11432, Saudi Arabia

⁴

School of Quantitative Sciences, Universiti Utara Malaysia (UUM), Sintok 06010, Kedah, Malaysia

⁵

Faculty of Education and Arts, Sohar University, Sohar 311, Oman

^*

Author to whom correspondence should be addressed.

Mathematics 2025, 13(17), 2766; https://doi.org/10.3390/math13172766

Submission received: 9 July 2025 / Revised: 8 August 2025 / Accepted: 22 August 2025 / Published: 28 August 2025

(This article belongs to the Special Issue Optimization Algorithms, Distributed Computing and Intelligence)

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Abstract

Nonlinear least squares (NLS) models are extensively used as optimization frameworks in various scientific and engineering disciplines. This work proposes a novel structured conjugate gradient (SCG) method that incorporates a structured diagonal approximation for the second-order term of the Hessian, particularly designed for solving NLS problems. In addition, an acceleration scheme for the SCG method is proposed and analyzed. The global convergence properties of the proposed method are rigorously established under specific assumptions. Numerical experiments were conducted on large-scale NLS benchmark problems to evaluate the performance of the method. The outcome of these experiments indicates that the proposed method outperforms other approaches using the established performance metrics. Moreover, the developed approach is utilized to address the inverse kinematics challenge in controlling the motion of a robotic system with four degrees of freedom (4DOF).

Keywords: structured vector; nonlinear; sufficient descent; diagonal; inverse kinematics

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MDPI and ACS Style

Yunus, R.B.; Ben Ghorbal, A.; Zainuddin, N.; Ibrahim, S.M. An Accelerated Diagonally Structured CG Algorithm for Nonlinear Least Squares and Inverse Kinematics. Mathematics 2025, 13, 2766. https://doi.org/10.3390/math13172766

AMA Style

Yunus RB, Ben Ghorbal A, Zainuddin N, Ibrahim SM. An Accelerated Diagonally Structured CG Algorithm for Nonlinear Least Squares and Inverse Kinematics. Mathematics. 2025; 13(17):2766. https://doi.org/10.3390/math13172766

Chicago/Turabian Style

Yunus, Rabiu Bashir, Anis Ben Ghorbal, Nooraini Zainuddin, and Sulaiman Mohammed Ibrahim. 2025. "An Accelerated Diagonally Structured CG Algorithm for Nonlinear Least Squares and Inverse Kinematics" Mathematics 13, no. 17: 2766. https://doi.org/10.3390/math13172766

APA Style

Yunus, R. B., Ben Ghorbal, A., Zainuddin, N., & Ibrahim, S. M. (2025). An Accelerated Diagonally Structured CG Algorithm for Nonlinear Least Squares and Inverse Kinematics. Mathematics, 13(17), 2766. https://doi.org/10.3390/math13172766

Note that from the first issue of 2016, this journal uses article numbers instead of page numbers. See further details here.

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An Accelerated Diagonally Structured CG Algorithm for Nonlinear Least Squares and Inverse Kinematics

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