Parallel manipulators are practically confined to a dramatically small subregion of their workspace due to type II singularities. Around such configurations, the actuator forces tend to infinity and, consequently, the controllability is lost. Therefore, in recent years, growing attention is devoted to develop methods for parallel manipulators to pass through these singular positions while the required actuator efforts remain bounded and continuous. With this aim, in this paper the singularity-consistent payload locations are analytically studied. First, it is shown for three-degree-of-freedom planar parallel manipulators that, in general, the corresponding locus describes a circle in the end-effector plane, whereas it will be a straight line if the angular acceleration of the end-effector platform is prescribed to be zero at the singular configuration. Then, it is proved for spatial six-degree-of-freedom parallel manipulators that the locus of interest represents a quadric surface, whereas it degenerates to a plane if the end-effector platform is prescribed to be in pure translation. The developed payload placement method is outlined as four theorems and one corollary, and its effectiveness is demonstrated through numerical simulations. (C) 2015 Elsevier Ltd. All rights reserved.