Information about the position of entities is very valuable in many fields. People, animals, robots and sensors are some examples of entities that have been targeted as nodes of interest for localization purposes. Technical advances in ubiquitous computing and wireless communications properties are very valuable means to obtain localization information. This paper presents a novel range-free localization algorithm based on connectivity and motion (LACM). The core of the algorithm is an error function that measures the error of the obtained trajectories with respect to the localization solution space, a multi-dimensional space that encompasses all solutions that satisfy completely the constraints of a range-free localization problem. LACM is a centralized method that can be used standalone or as a refinement phase for other localization methods. Limited-memory Broyden---Fletcher---Goldfarb---Shanno, an unconstrained optimization algorithm, is the numerical method used to minimize the error function. The performance of LACM is validated both through extensive simulations with excellent results in scenarios with irregular communications and by transforming real Bluetooth connectivity traces into localization information.