This paper presents the background to and some initial results of an attempt to develop the basis for quantitative performance measures of robot behaviour. First, an example of a simple robot behaviour is used to motivate the need for a dynamical systems approach to the understanding and investigation of robot behaviour. The background and initial theoretical developments necessary for defining some appropriate quantitative measures are then presented. Finally, an example of the application of one of the techniques proposed is presented, together with a discussion of the practical difficulties involved and the future prospects of the presented approach.