The stability of integrators dealing with high order Differential Algebraic Equations (DAEs) is a major issue. The usual procedures give rise to instabilities that are not predicted by the usual linear analysis, rendering the common checks (developed for ODEs) unusable. The appearance of these difficult-to-explain and unexpected problems leads to methods that arise heavy numerical damping for avoiding them. This has the undesired consequences of lack of convergence of the methods, along with a need of smaller stepsizes. In this paper a new approach is presented. The algorithm presented here allows us to avoid the interference of the constraints in the integration, thus allowing the linear criteria to be applied. In order to do so, the integrator is applied to a set of instantaneous minimal coordinates that are obtained through the application of the null space. The new approach can be utilized along with any integration method. Some experiments using the Newmark method have been carried out, which validate the methodology and also show that the method behaves in a predictable way if one considers linear stability criteria.

The method of the lower deformation energy has been successfully used for the synthesis of mechanisms for quite a while. It has shown to be a versatile, yet powerful method for assisting in the design of mechanisms. Until now, most of the implementations of this method used the dimensions of the mechanism as the synthesis variables, which has some advantages and some drawbacks. For example, the assembly configuration is not taken into account in the optimization process, and this means that the same initial configuration is used when computing the deformed positions in each synthesis point. This translates into a reduction of the total search space. A possible solution to this problem is the use of a set of initial coordinates as variables for the synthesis, which has been successfully applied to other methods. This also has some additional advantages, such as the fact that any generated mechanism can be assembled. Another advantage is that the fixed joint locations are also included in the optimization at no additional cost. But the change from dimensions to initial coordinates means a reformulation of the optimization problem when using derivatives if one wants them to be analytically derived. This paper tackles this reformulation, along with a proper comparison of the use of both alternatives using sequential quadratic programming methods. In order to do so, some examples are developed and studied.