Comprehensive approach for the dimensional synthesis of a four-bar linkage based on path assessment and reformulating the error function

Highlights

• Optimal dimensional synthesis procedure for general case of the four-bar linkage.

• Methodology for synthesizing path and motion generation.

• Systematic detection and definition of conditions to avoid order defect.

• Error function includes the slope of the tangent to the curve at each point.

• Adequate identification of branches and circuits avoiding penalty functions.

Abstract

The optimal dimensional synthesis methodology proposed in this article does not follow the more common approaches of recent years, such as heuristic methods, but aims to perfect one of the earlier and efficient approaches—gradient methods. The four-bar linkage in its general form is applied, analyzing crank and rocker input, ensuring the nonexistence of circuit and order defects. The procedure for obtaining the optimal mechanism is based on a characterization of the circuits and branches of the path traced by the coupler point, establishing the corresponding actuation mode based on a series of indices that avoid the use of penalty functions. The proposed optimization methodology defines a new error function that includes an additional characteristic, the slope of the tangent to each point of the path. Including the tangent enables better fitting to the desired curve without sacrificing the possibility of calculating the gradient analytically, and obtaining, in general, better results than by defining additional points. Although this paper focuses primarily on the dimensional synthesis of path generation, the bases of the procedure have been also applied to rigid body motion, as shown in the illustrative examples.

Introduction

Dimensional synthesis is a recurring and highly relevant theme in the mechanism theory literature, and many articles have been written on this topic. The earliest approach, and perhaps the most frequently used, is based on using gradient methods to minimize the squared error of distances. Currently, these methods are less used than more recent heuristic methods that have become more and more popular. This tendency can be observed in [1,2], which provide a chronological review of the evolution of dimensional optimization over the last 70 years. However, it seems strange that more effort has not been made to maximize the capabilities of earlier methods before focusing on newer techniques that are often more complicated. This article proposes an optimal synthesis methodology based on gradient methods, but incorporating various innovations. The four-bar linkage was chosen as the basis of these innovations because it best combines the characteristics of simplicity and capability to realize them.

When the number of precision points is limited, the dimensional synthesis can be solved by graphical or analytical methods, and an exact solution can be obtained. This is not possible with a large number of points. In the four-bar linkage, assuming the path generation case, the limit is nine points [3]. The first dimensional optimization methods consisted of applying a null gradient condition in an effort to minimize an error function that quantified the difference between the movement generated by the mechanism and the desired movement. This resulted in a nonlinear system of equations that included coefficients depending on passive variables, which generally could not be eliminated. For this reason, it was common to linearize the error function and solve the problem using iterative methods, starting from an approximate solution, so that the method would converge to the closest minimum (local optimization). In fact, this was the procedure used by Chi Yeh [4], the pioneer of computer-based dimensional optimization in the 1960s, who used as design variables the lengths of the bars and an error function based on the sum of squared distances. Another way to approach dimensional optimization using an analytical gradient is presented in [5], where the authors worked with the Cartesian coordinates of the points corresponding to the four revolute joints of the linkage, thus defining the configuration of the mechanism. The purpose of this approach, which is notably more complex, is to convert the original constrained dimensional optimization problem into an unconstrained problem. Later, the same authors expanded this idea by extending this method to the case of dimensional synthesis of rigid body motion [6]. In both [5] and [6], the authors focus solely on the four-bar linkage. The basics of these methods are applicable in principle to any mechanism. However, the prospect of extending them to more complex mechanisms is not simple because this requires obtaining analytically the partial derivatives of the synthesis equations with respect to the design parameters, which can be difficult or impossible in some cases. Faced with this situation, it is possible to use numerical differentiation, although this would markedly increase the computational complexity of the method and could degrade its accuracy relative to the exact gradient method. Other methods have also been proposed to obtain the gradient in an analytical-numerical form [7], where the passive variables are calculated by the Newton-Raphson method using the loop closure equations. Another possible solution, which has been proposed by the authors of this article and which will be described later, consists of implicitly differentiating the loop equations to obtain the partial derivatives of the passive variables. Finally, although they have not been extensively used, there are several local optimization methods that are not based on gradients and that do not require the use of derivatives, such as Powell’s method, which has also been used by various authors to address synthesis problems [8].

On the other hand, reference [9] presents a general approach for an arbitrary mechanism based on the gradient method. In that article, the synthesis and constraint equations are processed under the same conditions. For this purpose, the authors use natural coordinates, which facilitate a general formulation. Nevertheless, the same authors point out complications in the iterative process if multiple points are used, recommending that the synthesis be performed in several steps, which is a major inconvenience compared to other methods that consider all the points at once. It should be noted that this approach [9], as well as that previously mentioned in [5], does not in principle present any limitations for working with rocker-type input mechanisms, despite the fact that no methodology is implemented to avoid circuit and order defects. Later, Sancibrián et al. [10] pursued an idea similar to that presented in [9], but introduced several improvements. Among these, the following should be noted: the use of mixed instead of natural coordinates; the addition of a previous translation, rotation, and scaling phase; the imposition of conditions to avoid indeterminate positions; and the use of SQP (sequential quadratic programming) for optimization, which expands the error function to a quadratic form rather than the linear form that is usual in other gradient methods. Finally, concerning methods that are applicable to any mechanism, the energy method of Avilés et al. [11] (that includes the first author of this paper) must be mentioned, which is based on a second-order Newton optimization method and has the peculiar property that the function to be minimized is based on the elastic energy of the mechanism, assuming that the elements are deformable.

Another group of publications describes heuristic methods. One significant drawback of these methods is that they do not guarantee convergence to an optimal solution. Besides, their operation is influenced by a large number of parameters that can be difficult to determine beforehand and that can vary for each problem. For example, in the case of genetic algorithms, the parameters include the number of generations, population size, crossover and mutation rates, etc. Moreover, their computational complexity is higher than that of gradient methods, given that they must evaluate the objective function many more times. However, they offer some interesting advantages relative to the gradient method. First, they do not require derivative calculations or initial approximations, which normally have to be assumed by the designer. In specific cases, to obtain the initial approximation, curve atlases exist and can be consulted [12,13]. In general cases, trial and error is probably the best option. The reader should be aware that in gradient methods, the choice of an initial design is a critically important step and will have an important effect on the quality of the solution obtained by local methods. On the contrary, heuristic methods direct their search towards the global optimum, but in practice, they lead only to reasonably good results. For all these reasons, these methods are currently the most popular for synthesizing mechanisms. Many of them are based on nature, the most popular being genetic algorithms [14], [15], [16] and differential evolution [17,18,20]. Both of these are based on the evolution of a population where the individuals (possible solutions) undergo crossover, mutation, and selection operations and evolve in such a way that each generation of individuals is better adapted (reduces the error function). However, the two involve slight differences in implementation [21]. Perhaps the most important of these differences is that, whereas genetic algorithms use crossover to orient and direct themselves towards better solutions in the search space, differential evolution mainly uses mutation to play this role. Another approach that has been successfully applied to dimensional synthesis problems is the cuckoo method [22,23], which is based on the parasitic breeding behavior of certain birds. Along similar lines, the Modified Krill Herd [24], a later version of the Krill Herd algorithm [25], is based on the ability of these crustaceans to find food. Besides these, many other methods inspired by swarm behavior have been applied to synthesis, like Ant Search [26,27], which is based on the trail of pheromones left by ants during their search for food. In any case, not all algorithms are inspired by nature, but also by human social evolution, as is the case with the Imperialistic Competitive Algorithm [28,29], where the possible solutions are countries that form empires and acquire colonies depending on the power of each. Within the long list of existing heuristic methods based on the most diverse analogies, some researchers in the mechanism field have made their own modifications to improve algorithm functionality for dimensional synthesis. Such is the case with MUMSA (Malaga University Mechanism Synthesis Algorithm) [30], which is a slightly modified differential evolution algorithm. Other researchers have chosen to combine different methods, the results generally being known as hybrid algorithms. For example, reference [31] proposed an evolutionary method that combined differential evolution with a genetic algorithm. Similarly, Ref. [32] presents a new search procedure to obtain the best linkage by means of hybridizing a local search approach and an evolutionary algorithm. The work presented in [33] combines the Taguchi method, which was originally conceived for experimental design, with the Random Coordinate Search Algorithm. Another interesting idea is to use heuristic methods first to obtain an initial solution and then to continue with a gradient method [34]. The authors of [35] compared several of these techniques (genetic algorithms, differential evolution, and particle swarm) and observed that the quality of the solutions obtained was very similar, although the rate of convergence varied.

Finally, another group of methods can be distinguished that use search and comparison along with other mechanisms previously stored in a data base. In these cases, the fundamental concept is to describe the path to be generated using a number of pre-determined characteristics and afterwards to compare the result with the stored solutions. There are various ways to characterize these curves, among them, for example, the coupler angle function curve [36], which relates the orientation of the coupler with the input angle. Techniques based on basic splines and control polygons [37] have also been used to find the best solution in a data base of closed curves, taking into account the influences of translation, rotation, and scaling. Another common technique is to define paths using the coefficients corresponding with the first harmonics of a Fourier series and to compare them with those of the curve generated by the mechanism [38,39]. Later, some studies went further, successfully applying other common signal processing transformations, such as the Haar wavelet transform [40], which mainly differs from the Fourier transform because it contains not only frequency information, but also temporal information. It should also be mentioned that curve characterization using Fourier descriptors has been applied not only to data base search, but also for new objective function formulations used for optimization with genetic algorithms [15] or neural networks [41]. In fact, more and more researchers are proposing alternatives to the classical error function based on the sum of squared distances in an effort to resolve some of its problems, such as its sensitivity to translation, rotation, and scaling. Examples of novel objective functions are the circular proximity function [18,19], which measure the similarity of a group of points to a circular curve and have been successfully used in differential evolution. The approach presented in [19] is valid not only for the four-bar mechanism but for all four-link planar mechanisms. Also, in relation to other new objective functions, the turning function [42] characterizes the path in terms of a series of tangent angles at points along the curve.

After this review of existing optimization methods for dimensional synthesis of path generation and rigid body guidance, this paper will proceed to describe the novel elements implemented in the methodology proposed here. These elements will be discussed in the rest of this article, finishing with a series of examples that demonstrate the validity and effectiveness of the procedure. The novelties of this work are:

• The classification proposed by Ma and Ángeles in [43] is completed in relation to specific cases of branches and circuits in the four-bar linkage.

• Formulation of the possible cases using indices, which by its very definition avoids the occurrence of circuit defects without requiring the use of penalty functions.

• Systematic detection and definition of conditions to avoid order defect, particularly in the case of rocker input, where the condition to be verified for the input parameters is not as evident as in the case of crank input.

• Modification of the gradient method for the case of rocker input to prevent the occurrence of input values that exceed the range of motion of the mechanism, without the need to impose constraints of any type.

• Reformulation of the error function by including as characteristics, in addition to the coordinates of the prescribed points, the slope of the tangent to the curve at each of these points. This approach will make it possible to define the curve adequately with a smaller number of specified characteristics, thus facilitating the minimization problem.

• An alternative methodology to obtain the partial derivatives of the synthesis equations with respect to the passive variables of the mechanism, based on the implicit differentiation of the loop equations.

The proposed procedure is valid for the synthesis of path generation and for rigid body guidance. In fact, the examples presented include applications in both types of synthesis problems.

The paper is organized as follows: Section 2 establishes the necessary equations for the optimum synthesis and explains the unprescribed timing option. In Section 3, the analytical formulation of the branches and circuits of the coupler curve is established, as well as the way the proposed synthesis method approaches all possible situations. Section 4 presents the optimization methodology, including the formulation, the definition of the new error function including the slope of the curve and the algorithm flowchart. In Section 5, several examples demonstrate the validity of the methodology, and in Section 6, the computational data is included. Finally, in Section 7, the conclusions derived from this work are established.