Control and planning

This paper focuses on the problem of synthesizing an optimal four bar linkage of the type manivela-balancín for the generation of a required path. The synthesis of mechanisms for the generation of trajectories is currently an active area of research in which there are still many problems whose resolution is not entirely satisfactory. Complexity entailed in the optimization in this area is produced by the high nonlinearity of the equations obtained by translating the problem to a mathematical model.

Various optimization strategies proposed for synthesis of mechanisms are assessed by multiple numerical simulations on a series of curves type of coupler. The implementation is done in Matlab, making simulations on a conventional PC (Pentium 4 to 1700 Mhz).

He is carried out in the first place a preliminary analytical study of the mechanisms of the type crank and rocker, determining easily implemental conditions to ensure that the design space is reduced to those mechanisms that have guaranteed the rotabilidad and are characterized by optimal transmission angle values. Then carried out in this design space restricted a study of the role of error associated with the mechanism, which brings us to define different approaches for the calculation of the initial angle of the crank. Numerical simulations show that an approximation of the trigonometric type provides high quality solutions at a computational cost much less than the exhaustive search.

Finally, different optimization methods are implemented in the expanded design space. On the one hand, the previously proposed approach combined with a classic simple optimization method such as the dichotomous search. On the other hand, it implements a recently proposed method for global optimization based on a modification of the Lipschitziana approximation standard [1]. The numerical results show that the accuracy of the first technical proposal is more than sufficient for applications usual, requiring a time of computation on average 15 times lower than that of the Lipschitziana approach. As this last technique advantage arises not only the fact that allows for finding the global optimum more accurately, but also tolerated the incorporation of a number of variables, so it can also be used in the case in that it is found that determine the position of the fixed joints. Simulations show that, in the resolution of the problem full-time, the Lipschitziana approximation provides high quality solutions in reasonable computation times.

We conclude that the proposed methods of optimization, which do not require the objective calculation of the derivatives of the function and not even to have analytical expression, are particularly suitable for the synthesis of mechanisms, since that satisfactorily meet the objective of achieving an optimal mechanism with low computational cost.

[1] M. Björkman and k. Andersson, Global Optimization using the DIRECT algorithm in Matlab, Advanced Modeling and Optimization, 1 (2), (1999), pp. 9-16.

[3] L.W. Hi, Evolutions and Revolutions in Mechanical Engineering, Proc. XXV ASME Meeting, Phoenix, 1995