A gradient-based optimization method for designing linkages with velocity targets is described. Two theoretical application cases are established for four-bar linkage. In the first, a constant-velocity module is proposed for a point on the coupler. In the second, the goal is the velocity components. These cases are studied with and without coordination with the input link. The results obtained are compared with another gradient-based approach, and show that the method works efficiently for

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... types of target. used, by means of optimization techniques, both deterministic and stochastic. Among deterministic optimization methods, one of the few papers on dimensional synthesis with velocity targets is by Galeano et al. (6) . This article addresses the optimal design of a four-bar linkage with different goals, for which a coupler point must satisfy certain conditions for position and/or velocity. With respect to the velocity targets, they distinguish between a velocity target with implicit and explicit position targets. In the first case the aim is the Cartesian components of velocity. Thus, in addition to the velocity module, the path tangent is also imposed, which means that the shape of position is defined. In the second case the aim is a multiobjective optimization with both position and velocity targets. This interesting approach has an important drawback: it is only valid for four-bar linkage. This is because it is based on the analytical kinematic equations that define this mechanism. Obtaining these equations for a more complicated mechanism is not always possible. With respect to stochastic optimization methods, Fernández de Bustos et al. (7) deal with dimensional optimization problems with velocity targets. Using a multistage approach, they pose a global velocity target by means of Genetic Algorithms (GA). To compute the objective function, it is necessary to solve an optimization sub-problem in which position is the solution. Li et al. (8) propose the design of a press, which is powered by a hybrid system with a constant speed motor and servo motor, using GA with path and velocity targets. Stochastic methods, such as GA, can deal with global optimization problems. Nevertheless, they have a higher computational effort than deterministic. An optimal synthesis method is presented in this work. It has some advantages with respect to other synthesis methods, which can be summarized in: (a) the number of precision positions is not limited; (b) the computational effort is low, even compared with another well-known similar optimization method; and (c) it could be applied for a wide range of mechanisms. The authors have been developing a gradient-based local optimization method that can cope with traditional dimensional synthesis problems, namely path and function generation and rigid body guidance, in an efficient and simple way. This method is based on the analytical computation of Jacobian matrix of the objective function. This means that computational effort is reduced and the robustness of algorithm is increased, with respect to other gradientbased methods. In previous reports, the theoretical methodology is explained (9), (10) and some applications are described, such as design of a suspension system for road vehicles, Ackermann steering linkages and a feed mechanism for machinery (11), (12) . This paper further develops our previous work and introduces the formalism necessary to design linkages with velocity targets (13) . The application examples developed here are limited to the planar four-bar linkage, but more complex problems could be solved using the proposed approach. The systematic mathematical background is first described and then two different cases are presented. In order to contrast the proposed optimization method, a numerical comparison with a standard gradient-based approach is also developed.