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This paper presents the design process of a spherical robot capable of autonomous motion, and demonstrates how it can become a tool in child-development studies. The robot, named Roball, is capable of intentional self-propelled movements and can generate various interplay situations using motion, messages, sounds, illuminated parts and other sensors. Such capabilities allow Roball to interact with young children in simple and interesting ways, and to provide the potential of contributing to the development of their language, affective, motor, intellectual and social skills. Trials done with 12-24-month-old children demonstrate how Roball can be used to study children's interest in a self-propelled and intentional device. An experimental methodology to conduct such studies is presented: it is based on quantitative and qualitative techniques to evaluate interactions, thus enabling the identification of challenges and opportunities in child-robot interaction studies.