This paper investigates global uncertainty propagation and stochastic motion planning for the attitude kinematics of a rigid body. The Fokker–Planck equation on the special orthogonal group is numerically solved via noncommutative harmonic analysis to propagate a probability density function along flows of the attitude kinematics. Based on this, a stochastic optimal control problem is formulated to rotate a rigid body while avoiding obstacles within uncertain environments in an optimal fashion. The proposed intrinsic, geometric formulation does not require the common assumption that uncertainties are Gaussian or localized. It can be also applied to complex rotational maneuvers of a rigid body without singularities in a unified way. The desirable properties are illustrated by numerical examples.