This paper presents a new method to obtain an approximation of the field of movements of a 1-DOF linkage with lower pairs. The method is based on a linkage representation by natural coordinates and the storage of the constraint equations by means of a sparse cubic matrix. To obtain a discrete approximation of the field of movements, a three-stage process is used. In the first stage, a special Evolution Strategy is applied to make the population converges towards the zones where the constraints error is minimal, obtaining, at the same time, a good distribution of individuals. In the second stage, the final individuals of the ES are used as initial points for a derivative algorithm to obtain a greater accuracy. Later, the third stage is a filtering process to eliminate individuals that represent non-desired solutions. This method has been tested on simple linkages with well-known fields of movements, generating comprehensive outcomes that justify the validity of the method.