The study examines the application of a general minimum distance error function to the dimensional kinematic synthesis of bidimensional mechanisms. The minimum distance approach makes it possible to solve the problem maintaining the same generality as that of the minimum deformation energy method while solving the problems that occasionally appear in the former method involving low stiffness mechanisms. It is a general method that can deal both with unprescribed and prescribed timing problems, and is applicable for path generation problems, function generation, solid guidance, and any combination of the aforementioned requirements as introduced in the usual precision point scheme. The method exhibits good convergence and computational efficiency. The minimum distance error function is solved with a sequential quadratic programming (SQP) approach. In the study, the synthesis problem is also optimized by using SQP, and the function can be easily adapted to other methods such as genetic algorithms. In the study, the minimum distance approach is initially presented. Subsequently, an efficient SQP method is developed by using analytic derivatives for solving. The next point addresses the application of the concept for the synthesis of mechanisms by using an SQP approach with approximate derivatives. This delivers a situation where the optimization is performed on an error function that itself consists of an inner optimization function. A few examples are presented and are also compared with the minimum deformation energy method. Finally, a few conclusions and future studies are discussed.
The deformed energy method has shown to be a good option for dimensional synthesis of mechanisms. In this paper the introduction of some new features to such approach is proposed. First, constraints fixing dimensions of certain links are introduced in the error function of the synthesis problem. Second, requirements on distances between determinate nodes are included in the error function for the analysis of the deformed position problem. Both the overall synthesis error function and the inner analysis error function are optimized using a Sequential Quadratic Problem (SQP) approach. This also reduces the probability of branch or circuit defects. In the case of the inner function analytical derivatives are used, while in the synthesis optimization approximate derivatives have been introduced. Furthermore, constraints are analyzed under two formulations, the Euclidean distance and an alternative approach that uses the previous raised to the power of two. The latter approach is often used in kinematics, and simplifies the computation of derivatives. Some examples are provided to show the convergence order of the error function and the fulfilment of the constraints in both formulations studied under different topological situations or achieved energy levels.
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