With the number of diagnosed cases rising every year, Autism Spectrum Disorder (ASD) is in need of novel therapeutic approaches to counteract the social and motor impairments inflicted by it. In this research, one of such therapeutic approaches, movement therapy (in particular Dance Movement Therapy (DMT)), is combined with the concept of Socially Assistive Robots (SARs) to assess the viability of employing a new type of SAR. This SAR consists in the first ever Socially Assistive Drone (SAD): a quadrotor drone of reduced size conceived for maintaining human-drone interaction during therapeutic sessions meant for children diagnosed with ASD. The main focus of this paper consists in developing an adaptive control system based on Fuzzy Logic Control due to its inherent advantages such as intuitiveness or the ease with which expert knowledge can be introduced into control systems. In total, four different behavioural modes are developed for the SAD, together with an adaptive algorithm which influences both a set of adjustable parameters associated with each mode and the likelihood of it being selected based on the user specific preferences. The four behavioural modes and their respective adaptive algorithms are tested on 10 participants in 30-minute sessions. The variations in the adaptive parameters and the likelihood of each mode being selected for each individual are registered and reveal that the SAD is able to adapt its behaviour to each participant based on their respective levels of engagement.