Evaluation of the possibility of applying the deformable polyhedron method to the problem of optimization kinematic synthesis of a flat lever mechanism

Authors: Kotov A.V., Krol D. G., Ph. D. in Phys. And Math., Assoc. Prof.

This article is a translation of the original work of the same name, which was written in Russian and published in a peer-reviewed journal. I decided to prepare and publish its English version for several reasons. First, science and engineering thinking have no language barriers. Publishing a translation is a step towards drawing attention to my research from a wider audience, including foreign colleagues, engineers and researchers who may find the proposed method useful. Second, publishing the article in English helps increase the visibility of the blog itself in foreign search engines. This means that my developments and findings are more likely to reach those who truly need them. I am open to discussion, feedback and professional dialogue with anyone who finds the topic of my research relevant. I will be glad if this material proves useful beyond the Russian-speaking audience.

Introduction. In modern mechanical engineering, four-bar linkages are widely used due to their unique combination of design and functional advantages. These mechanisms, with a minimum number of links, make it possible to realize complex trajectories of the working parts of machines through the rational selection of their geometric parameters [1]. However, due to increased requirements for the efficiency, accuracy and reliability of lever mechanisms, traditional design methods based on experience and intuition are no longer sufficient to achieve the required kinematic parameters. Therefore, optimization kinematic synthesis of lever mechanisms is becoming a key factor ensuring the competitiveness of the equipment being developed.

In educational practice, the main focus is on geometric methods of synthesis of lever mechanisms, which are clear and relatively simple, but inferior in accuracy to the solution of the problem [2]. Recently, due to the widespread introduction of mathematical packages and programming languages, there has been a significant leap in the application of numerical optimization algorithms for kinematic synthesis of lever mechanisms [3 - 7]. As a result, geometric synthesis methods are gradually receding into the background, giving way to more accurate and efficient machine algorithms.

To date, there is no universal numerical algorithm capable of effectively solving the entire spectrum of optimization problems [8]. The application of popular gradient algorithms to the problems of optimization kinematic synthesis of lever mechanisms requires large computational resources and is not always effective. At the same time, the potential of gradient-free algorithms for solving this class of problems is not fully covered in the scientific literature, which, given their high adaptability for software implementation [9], requires additional research.

Aim of the research. To evaluate the possibility of applying a multi-parameter gradient-free optimization algorithm based on the deformed polyhedron method to solve the problem of optimization kinematic synthesis of a flat lever mechanism. To provide a qualitative assessment of the use of this method when it is implemented in the mathematical package PTC MathCAD.

Research methods. The analytical description of the kinematics of the flat lever mechanism under consideration is based on the vector method of coordinate transformation in an invariant basis. Numerical optimization methods and mathematical programming were used to perform the optimization kinematic synthesis of the mechanism.

Statement and solution of the kinematic analysis problem. Among four-bar linkages, a special place belongs to the so-called lambda mechanism (lambda-shaped mechanism or Chebyshev mechanism), the kinematic diagram of which is shown in Figure 1. With a certain combination of link lengths, the lambda mechanism converts the rotational motion of the driving link into an approximate rectilinear motion of one of its points over a certain limited section of its trajectory [10]. Despite the fact that today there is a tendency to gradually replace lambda mechanisms with more accurate mechanical systems, these mechanisms are still actively used in mechanical engineering [11, 12], and the evaluation of the effectiveness of many optimization algorithms is tested on this particular mechanism [13, 14].

Figure 1 – Kinematic diagram of the lambda mechanism:
1 – crank; 2 – connecting rod; 3 – rocker arm

Let for the considered lambda mechanism (see Figure 1), having arbitrary geometric parameters, it is required to provide the most straight-line section of the trajectory of the connecting rod curve of point M. This trajectory must be located at some given distance vertically YM, and pass through three points, each corresponding to a specific angle j of rotation of the driving link. Let us assume that two extreme points for j=90° and -90° are located at a distance ±XM horizontally relative to the origin of the adopted coordinate system, and the ordinate of the middle point for j=0° coincides with the origin of the coordinate system (see Figure 1).

The link lengths of the lever mechanism, as well as the coordinate of point O along the X axis, will act as optimized parameters, which we represent as the following vector of optimized parameters:

Before proceeding to solve the problem of optimization kinematic synthesis of the considered mechanism, it is necessary to describe the kinematics of motion of all its characteristic points taking into account the vector of optimized parameters (1) and the angle j of rotation of the driving link relative to the horizontal axis X of the adopted coordinate system (generalized coordinate). The analytical description of the kinematics of the lambda mechanism is based on the vector method of coordinate transformation in an invariant basis, presented in works [15 - 17]. Using the analytical dependencies given in these works, we describe the kinematics of the considered lever mechanism:

Expression (3) obtains the vector of link OA by rotating the unit vector of the X axis by an angle j counterclockwise (there is a «+» sign in front of the angle) with a change in its length to the optimized length p1ºLOA.

Expression (5) obtains the angle αBCA with vertex at point C using the law of cosines, using two optimized link lengths p2ºLBC and p3ºLAB, as well as the modulus of the vector of link CA, which determines the distance between points C and A.

Expression (8) obtains the vector of link BM, which is collinear to the original rotated vector of link AB (the rotation of the original vector is performed by zero angle).

The given expressions (2) - (9) are functions depending not only on the generalized coordinate – the angle j of rotation of the driving link, but also on the vector (1) of optimized parameters.

Statement and solution of the optimization synthesis problem. In most cases, the analytical solution of the optimization kinematic synthesis problem of a lever mechanism makes it possible to approximately implement the required trajectory of motion of the point of interest of the mechanism. However, due to the development and implementation of highly efficient optimization algorithms in modern mathematical packages and programming languages, it has become possible to significantly increase the accuracy of solving such a problem.

The task of any optimization synthesis is to minimize a certain objective function. As a rule, when forming such an objective function, the method of least squares of residuals of some calculated parameter from its optimal (permissible) value is used [5, 6]. For the considered mechanism, the deviation of the calculated trajectory of the connecting rod curve of the radius vector of point M from the given optimal trajectory of the radius vector of point M* will act as such a residual, and the formed objective function itself can be represented as follows:

where n is the number of specified points of the straight-line section of the trajectory of point M.

To carry out the optimization kinematic synthesis of the considered lambda mechanism by minimizing the formed objective function (10), the deformed polyhedron method, also known as the Nelder–Mead method [18], was applied. This method works with a simplex – a geometric figure, each vertex of which corresponds to a certain vector of a set of optimized parameters.

For the software implementation of the optimization algorithm of the deformed polyhedron, well-known block diagrams [8, 18, 19], examples of algorithm adaptation in various mathematical packages and programming languages [19 - 22], as well as works [23 - 25] were used. As a result, a fully functional optimization algorithm using the deformed polyhedron method [26] was implemented in software for the mathematical package PTC MathCAD.

The features of this algorithm included: the use of easily readable program code through the use of meaningful letter designations for simplex vertices (instead of i-th indices); automated formation of vertices of the initial regular simplex integrated into the algorithm body; a number of minor program improvements aimed at increasing the efficiency of the algorithm. It should also be noted that due to the discovered discrepancy in the published block diagrams of the algorithm, the original scheme [18] was adopted as the basis.

The performance of the optimization algorithm of the deformed polyhedron for the mathematical package PTC MathCAD was tested on known test functions [27]. As the research results showed, the deformed polyhedron method has a sufficiently high accuracy of the solution, as well as convergence speed due to the absence of operations with derivatives. As a result, this method can potentially allow the use of a large number of optimized parameters, as well as the application of more complex optimization criteria, which is especially important when carrying out optimization kinematic synthesis of multi-link lever mechanisms.

Results and discussion. From analytical geometry it is known that if it is necessary to construct a regular simplex in space, one of whose vertices is located at the point of the initial vector of optimized parameters, then the coordinates of the remaining vertices of such a simplex can be conveniently specified using an N×(N+1) matrix [8, 9]. For the adopted vector of optimized parameters (1), consisting of five elements, this matrix will have the form:

where a is the distance between two vertices of the simplex; N is the number of simplex vertices; pin is the initial vector of optimized parameters, used as the initial approximation in the optimization algorithm.

After the matrix of vertices of the initial regular simplex is formed, the proposed mathematical algorithm of optimization by the deformed polyhedron method [26] can be applied, for which the input parameters will be the matrix of vertices of the regular simplex (11) and the objective function (10), and the output parameter will be the vector of found optimized parameters:

Figure 2 shows the visualization in the mathematical package PTC MathCAD of the kinematic diagram of the considered lever mechanism before and after performing the optimization kinematic synthesis by the deformed polyhedron method, and the table shows the numerical values for the vector of initial and optimized parameters.

a)                                                                    b)
Figure 2 – Visualization in the mathematical package PTC.MathCAD of the kinematic diagram of the mechanism before (a) and after (b) the optimization synthesis

Table – Results of optimization synthesis, mm

As can be seen from the obtained calculation results (see Figure 2, b), the straight-line section of the trajectory of the connecting rod curve of point M fully satisfies all the stated requirements (passes through all specified points). At the same time, the values of three optimized parameters (link lengths LBC, LAB, LBM) are obtained practically equal to each other, which corresponds to the known analytical expressions [1, 10].

It should be noted that today all modern optimization algorithms must be able to solve problems taking into account functional constraints. The solution of such problems is of great practical importance for mechanical engineering, since in the process of kinematic synthesis, certain functional constraints (for example, restrictions on link lengths) can be imposed on all optimized parameters of lever mechanisms. As the research results showed, the optimization algorithm of the deformed polyhedron can be adapted to solve problems of conditional optimization with constraints in the form of equalities and/or inequalities without significantly reducing the accuracy and speed of the solution, using the method of penalty or barrier functions [8].

To solve the problem of optimization kinematic synthesis of a lever mechanism with constraints, all imposed constraints in the form of equalities or inequalities must be reduced to the following form:

Then, using the penalty function method, the total penalty functions can be represented as [8, 20]:

where Ch and Cg are penalty coefficients for equalities and inequalities.

After that, the formed general penalty function (12) must be added to the objective function (10) and the optimization problem must be solved again using the deformed polyhedron method:

Adequacy check. To verify the reliability of the found global minimum of the objective function, two most common approaches were used. The first approach consisted of comparing with the calculation results obtained using other already verified optimization algorithms, and the second – of comparing with the calculation results obtained for different initial values of the initial vector of optimized parameters.

The verification of the optimization algorithm using the first approach was carried out using the numerical minimization function built into the mathematical package PTC MathCAD [20, 22], the results of which are summarized in the table:

The verification using the second approach was carried out by changing the values for the vector of optimized parameters within ±30% of their initial values, followed by tracking the search for the optimal solution by the algorithm. All obtained verification results confirmed a sufficiently stable convergence of the optimization algorithm to the same global minimum of the objective function.

Conclusion. The paper presents an algorithm and the results of optimization kinematic synthesis of a lever mechanism using the deformed polyhedron method. This method showed a high speed of searching for an optimal solution, effective application with a large number of optimized parameters, as well as the possibility of setting the problem with or without additional constraints. The known sensitivity of the deformed polyhedron method to initial conditions manifests itself on strongly ravine functions, which must be taken into account when applying this optimization algorithm to a particular problem.

References

1. Artobolevskij, I. I. Mekhanizmy v sovremennoj tekhnike: sprav. posobie: v 7 t. / I. I. Artobolevskij – 2-e izd. pererab. – M. : Nauka, 1979. – T. 1 : Elementy mekhanizmov. Prostejshie rychazhnye i sharnirno-rychazhnye mekhanizmy. – 496 s.

2. Artobolevskij, I. I. Sintez ploskih mekhanizmov: ucheb. dlya vtuzov / I. I. Artobolevskij, N. I. Levitskij, C. A. Cherkudinov – M. : Fizmatgiz, 1959. – 1084 s.

3. Yao, X. Optimal synthesis of four-bar linkages for path generation using the individual repairing method / X. Yao, X. Wang, W. Sun et. al. // Mechanical Sciences. – 2022. – Vol. 13, no. 1. – P. 79–87.

4. Garcia-Marina, V. Optimum dimensional synthesis of planar mechanisms with geometric constraints / V. Garcia-Marina, I. Fernandez de Bustos, G. Urkullu et al. // Meccanica. – 2020. – Vol. 55. – P. 2135-2158.

5. Halturin, M. A. Sintez pryamolinejno-napravlyayushchego mekhanizma dlya otrezki zagotovok ehskimo / M. A. Halturin // Vestnik Vologodskogo gosudarstvennogo universiteta. Seriya: Tekhnicheskie nauki. – 2019. – № 1(3). – S. 27-34.

6. Bejsenov, N. K. Optimizacionno-metricheskij sintez sharnirnogo chetyrehzvennika / N. K. Bejsenov // Tekhnicheskie nauki - ot teorii k praktike. – 2016. – № 55. – S. 65-77.

7. Gebel', E. S. Optimizacionnyj kinematicheskij sintez chetyrehzvennogo rychazhnogo mekhanizma po dvum zadannym polozheniyam / E. S. Gebel', E. A. CHigrinova // Omskij nauchnyj vestnik. – 2020. – № 3(171). – S. 21-25.

8. Himmel'blau, D. Prikladnoe nelinejnoe programmirovanie / D. Himmel'blau; per. s angl. I. N. Byhovskoj i B. T. Vavilova ; pod red. M. L. Byhovskogo. – M. : Mir, 1975. – 534 s.

9. Attetkov, A. V. Metody optimizacii: ucheb. dlya vuzov / A. V. Attetkov, S. V. Galkin, V. S. Zarubin; pod. red. V. S. Zarubina, A. P. Krishchenko. – 2-e izd., stereotip. – M. : MGTU im. N.EH. Baumana, 2003. – 440 s.

10. Baranov, G. G. Kurs teorii mekhanizmov i mashin / G. G. Baranov. – M. : Mashinostroenie, 1975. – 494 s.

11. Budzilo, E. E. Ispol'zovanie novogo bul'dozernogo oborudovaniya s mekhanizmom Chebysheva pri proizvodstve stroitel'nyh rabot / E. E. Budzilo, E. V. Grechishkina, M. YU. Psyuk // Naukoemkie tekhnologii i oborudovanie v promyshlennosti i stroitel'stve. – 2023. – № 2(76). – S. 106-113.

12. Manickavelan, K. Design, Fabrication and Analysis of Four Bar Walking Machine Based on Chebyshev’s Parallel Motion Mechanism / K. Manickavelan, B. Singh, N. Sellappan // European International Journal of Science and Technology. – 2014. – Vol. 3, no. 8. – P. 65–73.

13. Mehdigholi, H. Optimization of Watt's Six-Bar Linkage to Generate Straight and Parallel Leg Motion / H. Mehdigholi, S. Akbarnejad // International Journal of Advanced Robotic Systems. – 2012. – Vol. 9, no. 22. – P. 1–6.

14. Polyakov, B. N. Optimizaciya kinematicheskih parametrov rychazhnyh chetyrehzvennyh mekhanizmov / B. N. Polyakov // Prikladnaya informatika. – 2010. – № 3(27). – S. 108-112.

15. Bobirenko, S. N. Modelirovanie processa raboty mekhanizma podpressovki pitayushchego apparata kormouborochnogo kombajna / S. N. Bobirenko, A. V. Kotov // Vestnik Belorussko-Rossijskogo universiteta – Mogilev, 2011. №1 (30). – S. 18-26.

16. Kotov, A. V. Optimizaciya parametrov predohranitel'nogo ehlementa pal'chikovogo mekhanizma shneka zhatki zernouborochnogo kombajna / A. V. Kotov // Traktory i sel'hozmashiny. – 2023. – T. 90. – №1. – C. 13–24.

17. Kotov, A. V. Analiz uravnoveshennosti krivoshipno-polzunnogo mekhanizma privoda rezhushchego apparata metodom vektorov glavnyh tochek / A. V. Kotov // Traktory i sel'hozmashiny. – 2024. -– T. 91. – №2. – C. 167–180.

18. Nelder, J. A. A simplex method for function minimization / J. A. Nelder, R. Mead // Computer joornal. – 1965. – Vol. 7. – P. 308–313.

19. Bandi, B. Metody optimizacii: vvodnyj kurs / B. Bandi; per. s angl. O. V. ShihEEvoj; pod red. V. A. Volynskogo. – M. : Radio i svyaz', 1988. – 128 s.

20. Gal'chenko, V. YA. MathCAD: matematicheskie metody i instrumental'nye sredstva optimizacii / V. YА. Gal'chenko, R. V. Tremboveckaya. – CHerkassy: CHP Gordienko E. I., 2018. – 516 s.

21. MAt'yuz, Dzhon G. CHislennye metody. Ispol'zovanie MATHLAB / Dzhon G. MAt'yuz, Kurtis, D. Fink – 3-e izdanie. : per. s angl. – M. : Vil'yams, 2001. – 720 s.

22. Dmitrieva, T. L. Realizaciya uslovnoj zadachi nelinejnogo matematicheskogo programmirovaniya s ispol'zovan'metoda deformiruemogo mnogogrannika v programme MathCAD / T. L. Dmitrieva, V. T. Nguen // Sovremennye tekhnologii. Sistemnyj analiz. Modelirovanie. – 2014. – № 4(44). – S. 73-79.

23. Lagarias, J. C. Convergence of the restricted Nelder-Mead algorithm in two dimensions / J. C. Lagarias, B. Poonen, M. H. Wright // SIAM Journal on Optimization. – 2012. – Vol. 22, no. 2. – P. 501–532.

24. Sufiyanov, V. G. Metod Neldera-Mida resheniya zadachi optimizacii geometricheskoj formy stvola avtomaticheskoj pushki dlya uluchsheniya kolebatel'nyh harakteristik / V. G. Sufiyanov, D. A. Klyukin, I. G. Rusyak // Izvestiya Samarskogo nauchnogo centra Rossijskoj akademii nauk. – 2023. – T. 25, №4(114). – S. 121-131.

25. Petrushin, A. D. Optimizaciya aktivnoj chasti ventil'no-induktornogo ehlektrodvigatelya / A. D. Petrushin, A. V. Kashuba // Vestnik Rostovskogo gosudarstvennogo universiteta putej soobshcheniya. – 2016. – № 1(61). – S. 61-65.

26. Algoritm deformiruemogo mnogogrannika [Electronic resource]. – Access mode : https://drive.google.com/file/d/172ozZXeprDCfWc3eA4_-xJd_zgVEIpyv. – Date of access: 01.11.2025.

27. Timofeeva, O. P. Issledovanie populyacionnyh algoritmov v reshenii zadach nepreryvnoj optimizacii / O. P. Timofeeva, S. A. Neimushchev, L. I. Neimushcheva // Trudy NGTU im. R.E. Alekseeva. – 2018. – № 4(123). – S. 48-55.

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To cite this work:

Котов, А. В. Оценка возможности применения метода деформируемого многогранника к задаче оптимизационного кинематического синтеза плоского рычажного механизма / А. В. Котов, Д. Г. Кроль // Механика. Исследования и инновации. – 2025. – № 18. – С. 81-89. – EDN KIRXMS.

Kotov A. V., Krol D. G. Ocenka vozmozhnosti primeneniya metoda deformiruemogo mnogogrannika k zadache optimizacionnogo kinematicheskogo sinteza ploskogo rychazhnogo mekhanizma [Evaluation of the possibility of applying the deformable polyhedron method to the problem of optimization kinematic synthesis of a flat lever mechanism]. Mekhanika. Issledovaniya i innovacii [Mechanics. Researches and innovations], 2025, no. 18, pp. 81–89 (in Russ.).

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