  In this article we investigate the phase transition phenomena that occur in a
model of self-organisation through body-attitude coordination. Here, the
body-attitude of an agent is modelled by a rotation matrix in $\mathbb{R}^3$ as
in [Degond, Frouvelle, Merino-Aceituno, 2017]. The starting point of this study
is a BGK equation modelling the evolution of the distribution function of the
system at a kinetic level. The main novelty of this work is to show that in the
spatially homogeneous case, self-organisation may appear or not depending on
the local density of agents involved. We first exhibit a connection between
body-orientation models and models of nematic alignment of polymers in higher
dimensional space from which we deduce the complete description of the possible
equilibria Then, thanks to a gradient-flow structure specific to this BGK
model, we are able to prove the stability and the convergence towards the
equilibria in the different regimes. We then derive the macroscopic models
associated to the stable equilibria in the spirit of [Degond, Frouvelle,
Merino-Aceituno, 2017] and [Degond, Frouvelle, Liu, 2015].
