Method of inverse 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

The kinematic synthesis of a planar lever mechanism consists in determining the parameters of its kinematic scheme according to the specified motion conditions of the driven link. One of the varieties of this problem is the synthesis of a mechanism that reproduces a given trajectory of motion of a certain link or point [1]. Such mechanisms are still in demand in many branches of mechanical engineering [2 - 4], and their optimal design is directly related to the formulation and solution of the kinematic synthesis problem.

Despite the continuous development and improvement of methods for kinematic synthesis of planar lever mechanisms, there is no unified approach to solving this problem. Known graphical methods [5] are effective only for the simplest planar lever mechanisms and are practically inapplicable for multi-link mechanisms. Grapho-analytical methods [6] of kinematic synthesis are developed for solving problems for specific lever mechanisms and, as a rule, do not possess high flexibility in solving other problems. Known analytical methods [7 - 9] of kinematic synthesis of multi-link lever mechanisms are in the vast majority quite complex, cumbersome, and may use optimization algorithms unavailable for detailed study. Therefore, the development of new, visual, and universal methods for kinematic synthesis of planar lever mechanisms capable of easy adaptation in specialized mathematical packages and programming languages still represents an important scientific and practical task.

Figure 1 – Kinematic scheme of a planar lever mechanism

Traditionally, when solving the problem of kinematic synthesis for reproducing a given trajectory of motion of a point of a planar lever mechanism, the direct problem of kinematics is preliminarily solved, based on the results of which a certain objective function is formed. As a rule, such an objective function is the residual of the coordinates of the obtained calculated trajectory of the coupler curve of a certain point with the required trajectory [10] or their standard deviation [11]. There are works in which the objective function is a function that simultaneously takes into account the squares of the differences between the areas of the described regions and the lengths of the desired and current trajectories [12]. Minimization of the formed objective function is achieved using one or another optimization algorithm by selecting rational lengths of the moving links, and in some cases, the coordinates of the frame dimensions. The classical methods used to solve the direct problem of kinematics are in the vast majority based on the method of closed vector contours by Zinoviev [13, 14] and require significant mathematical calculations. But, as practice shows [15], the fewer analytical expressions are embedded in the mathematical model of the mechanism, the more efficiently the search for an optimal solution is carried out during its kinematic synthesis.

This paper proposes a method for solving the inverse problem of kinematics of a planar lever mechanism, using directly the coordinates of the target points of the trajectory for the specified design positions. At the same time, the minimum number of analytical expressions is achieved through the use of the method of coordinate transformation in a fixed basis using the theory of complex numbers, and as the objective function it is proposed to use the standard deviation of the length of only one of the links of the mechanism. As a result of such formulation and solution of the problem of reproducing a given trajectory of motion of a point, an effective method with acceptable accuracy and speed of finding an optimal solution is obtained.

The aim of this study is to develop a method for solving the problem of reproducing a given trajectory of motion of a point of a planar lever mechanism by the method of inverse kinematic synthesis taking into account imposed functional constraints. Computer implementation of the proposed method is presented in the PTC MathCAD mathematical package.

Materials and Methods

The kinematic analysis of the planar lever mechanism is based on the method of coordinate transformation in a fixed basis using the theory of complex numbers. The solution of the inverse kinematic synthesis problem is carried out by a numerical method based on the built-in function minimize() of the PTC MathCAD mathematical package using the quasi-Newton method.

Method of Inverse Kinematic Synthesis

Let us consider the formulation and solution of the problem by the proposed method of inverse kinematic synthesis using the example of a planar four-bar linkage, the kinematic scheme of which, corresponding to one of the possible assemblies, is shown in Figure 1. The driving link (crank) is link AC, and the trajectory of motion of point M required for reproduction is specified by the coordinates for the n-th number of target points on the complex plane (in the Re-Im coordinate system).

When performing kinematic synthesis, as the optimized parameters we choose all the lengths of the links of the mechanism (except the driving link) and the coordinates of the fixed points, which we represent in the form of the following vector of independent optimized parameters:

In the general case, the given trajectory of motion of point M can be any curve. For testing and debugging the proposed method of inverse kinematic synthesis, we assume that for the considered planar lever mechanism the required trajectory of motion of point M has the form of an ellipse, the major axis of which is parallel to the real axis of the adopted coordinate system, and is specified using the following 10 target points [16 - 18]:

The solution of the inverse problem of kinematics for the considered lever mechanism will be carried out by the method of coordinate transformation in a fixed basis using the theory of complex numbers [19, 20]. This method, in addition to its simplicity and clarity, has another undeniable advantage – the minimum number of analytical expressions, which, as noted above, is a fairly weighty argument when solving any problem of kinematic synthesis.

Using the analytical dependencies given in works [19, 20], as well as the vector of optimized parameters (1), we describe the kinematics of the considered lever mechanism. To do this, we represent the intermediate calculation parameters (coordinates of points, link lengths, etc.) in the form of a radius vector of a complex number, for the designation of which hereinafter the underscore symbol will be used. At the same time, when solving the inverse problem of kinematics, no generalized coordinate will be used, since only n static positions of the mechanism will participate in the analytical description.

Points M, A, and B:

where j is the imaginary unit.

Link BM:

The angle MBD is found by the law of cosines using the user-defined function Tcos() [19, 20]:

The vector of link BD is found using the user-defined function TurnRI() by rotating the found radius vector of link BM by the angle aMBD clockwise (the angle is preceded by a minus sign) with a change in its length to the optimized length p4ºLBD [19, 20]:

For brevity, we give below only the calculation formulas for the remaining parameters [19, 20]:

As a result, the inverse problem of kinematics of the considered planar lever mechanism has been solved. At the same time, all found quantities depend on the vector of optimized parameters and are determined for a specific position of the radius vector of point M.

The further solution of the inverse kinematic synthesis problem consists in forming the objective function, which is the central element in solving any synthesis problem. When solving the problem under consideration, the residual of the coordinates of the trajectories of point M can no longer act as the objective function, since this trajectory was taken as the basis when describing the kinematics of the mechanism. In the proposed method of inverse kinematic synthesis, the standard deviation of the length of the closing link AC in all design positions will act as the objective function:

Such an approach to solving the kinematic synthesis problem reduces to minimizing the objective function of only one link (the closing one), which makes the solution of the problem simpler and more visual. The closer to zero during kinematic synthesis the value of the objective function (2) is obtained, the more constant the length of the closing link will be for all considered positions of the mechanism, and, consequently, the minimum deviation of point M from the given trajectory will be ensured.

It should also be noted that when solving the inverse problem of kinematic analysis, the driving link does not necessarily have to act as the closing link for forming the objective function. Any other link convenient for the analytical description of the inverse problem of kinematics of the considered mechanism can be chosen as the closing link.

For the correct operation of the formed objective function (2), certain functional constraints must be imposed on the search for a solution to the inverse kinematic synthesis problem. For this purpose, all lengths of the links of the considered mechanism, as well as the coordinates of its fixed joints used in the vector of optimized parameters (1), are limited to a certain search range, which in practice, as a rule, is established based on certain requirements for the layout of the mechanism. This search range is adopted by analogy with works [16 - 18], increasing it by 10 mm relative to the values adopted in them:

In addition, we ensure the condition for the existence of a triangle for the rigid link MDC, according to which a triangle exists if and only if the sum of any two of its sides is greater than the third side:

To ensure the condition for the existence of a crank for the considered four-bar linkage, we use Grashof's theorem:

Using the second equation in expression (5), it is determined that the input link AC is the smallest. At the same time, the length of link AC is set equal to the average value in all design positions:

After the objective function (2) has been formed and functional constraints (3) – (5) have been specified, the inverse kinematic synthesis problem of the mechanism reduces to the classical search for the values of the vector of optimized parameters (1) at which it reaches a minimum with an acceptable value. The acceptable minimum value corresponds to permissible deviations of the link lengths and coordinates of the fixed joints, and, consequently, to acceptable accuracy of reproducing the required trajectory of motion of point M.

The search for the minimum value of the objective function can be carried out using one or another optimization algorithm (for example, the deformable polyhedron method [15], genetic [21], etc.) or using special built-in functions in mathematical packages or programming languages. To date, the development of numerical methods and optimization algorithms using modern mathematical packages and programming languages allows solving the problems of kinematic synthesis of planar lever mechanisms at a qualitatively new level.

In this work, the numerical solution of the inverse kinematic synthesis problem will be carried out in the PTC MathCAD mathematical package using the built-in function minimize(f, var) [22]. This function can search for the extremum of the objective function by the conjugate gradient method or the quasi-Newton method with acceptable accuracy and in a short time. The software implementation of the search for the minimum of the objective function in the PTC MathCAD mathematical package is shown in Figure 2 and, for simplicity of its description, is divided into five conditional blocks.

In the first block, the objective function and the average length of the closing link are specified, which were calculated for all given positions. In the second block, for the correct operation of the algorithm, the initial approximation for all quantities of the vector of optimized parameters is recorded. In the third block, in the form of a system of inequalities, the ranges of permissible values of the optimized parameters (left part of the block) are specified, as well as the adopted functional constraints (right part of the block). In the fourth block, the inverse kinematic synthesis problem itself is solved, taking into account all the functional constraints specified above, with the final output of the found synthesis results. In the fifth block, the fulfillment of all specified functional constraints is checked, and the values of the objective function and the length of the closing link for the found optimal kinematic parameters of the mechanism are output.

Figure 2 – Software implementation of minimizing the objective function in the PTC MathCAD mathematical package

Research Results

As a result of solving the inverse kinematic synthesis problem, the values of the vector of optimized parameters and the average value of the length of the closing link AC were obtained (see Figure 2). Using these values, the final trajectory of point M was refined by performing a repeated kinematic analysis, but in the forward order using the generalized coordinate (rotation angle φ of the driving link) [19, 20]:

Figure 3, a shows the visualization in the PTC MathCAD mathematical package of the kinematic scheme of the synthesized mechanism [23, 24], and Figure 3, b shows the graph of the obtained trajectory of motion of point M*, passing through the given target points.

a – kinematic scheme of the synthesized mechanism; b – trajectory of motion of point M*
Figure 3 – Visualization in the PTC MathCAD mathematical package of the kinematic scheme of the synthesized planar lever mechanism and the trajectory of motion of point M*

The estimation of the deviation error of the obtained trajectory of motion of point M* from the given target points will be carried out using radius vectors. As the reference point of the radius vector, point O, we take the center of gravity of the flat figure (ellipse, see Figure 3, b) bounded by the given points, and determined using the expression:

The compared radius vectors OMi (initial data) and OM* (calculated data) are obtained by the formulas:

Here it should be noted that for correct comparison of radius vectors (7) with each other, they must have the same angle of inclination to the real axis of the complex plane, which can be found from the argument of the corresponding complex number vector:

Then the absolute and relative error of deviation of the obtained trajectory of motion of point M* at the given target points is calculated as:

The graphs of the calculated absolute and relative errors at each target point of the trajectory of motion of point M are shown in Figure 4.

a                                                        b
a – absolute error graph; b – relative error graph
Figure 4 – Graphs of changes in absolute and relative errors at the target points of the trajectory of motion of point M

As can be seen from the obtained calculation results, the maximum absolute error of deviation of the trajectory of point M* at the control points does not exceed 0.1 mm in absolute units or 1% in relative units. This indicates a fairly high accuracy of the calculated values of the vector of optimized parameters, and also confirms the correctness of the formulation and solution of the obtained inverse kinematic synthesis problem.

Conclusion

The proposed method of inverse kinematic synthesis of a planar lever mechanism allowed for approximate reproduction of a given trajectory of motion of a point of the mechanism with fairly high accuracy (the calculation error at the target points did not exceed 1% in relative units). The proposed method is based on solving the inverse problem of kinematic synthesis with subsequent formation of the objective function in the form of the standard deviation of the length of one closing link.

This method showed high adaptation in the PTC MathCAD mathematical package, however, the accuracy of the method will entirely depend on the capabilities of the optimization algorithm used, capable of finding the global minimum of the objective function taking into account the imposed constraints.

The results of the work can be used in practice in the design of new or modernization of existing lever mechanisms in various branches of mechanical engineering.

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To cite this work:

Котов А. В. Метод обратного кинематического синтеза плоского рычажного механизма / А. В. Котов, Д. Г. Кроль // Вестник Брестского государственного технического университета. – 2026. – № 2(140). – С. 78-83. – DOI: 10.36773/1818-1112-2026-140-2-78-83.

Kotov A. V., Krol D. G. Metod obratnogo kinematicheskogo sinteza ploskogo rychazhnogo mekhanizma [Method of inverse kinematic synthesis of a flat lever mechanism]. Vestnik Brestskogo gosudarstvennogo tekhnicheskogo universiteta [Vestnik of Brest State Technical University], 2026, no. 2(140), pp. 78-83. DOI: https://doi.org/10.36773/1818-1112-2026-140-2-78-83 (in Russ.).

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