Analysis of the degrees-of-freedom of spatial parallel manipulators in regular and singular configurations R. Arun Srivatsan a , Sandipan Bandyopadhyay a, ⁎, Ashitava Ghosal b a Department of Engineering Design, Indian Institute of Technology Madras, India b Department of Mechanical Engineering, Indian Institute of Science, Bangalore, India article info abstract Article history: Received 9 September 2012 Received in revised form 16 April 2013 Accepted 26 April 2013 Available online 20 June 2013 This paper presents a study of the nature of the degrees-of-freedom of spatial manipulators based on the concept of partition of degrees-of-freedom. In particular, the partitioning of degrees- of-freedom is studied in five lower-mobility spatial parallel manipulators possessing different combinations of degrees-of-freedom. An extension of the existing theory is introduced so as to analyse the nature of the gained degree(s)-of-freedom at a gain-type singularity. The gain of one- and two-degrees-of-freedom is analysed in several well-studied, as well as newly developed manipulators. The formulations also present a basis for the analysis of the velocity kinematics of manipulators of any architecture. © 2013 Elsevier Ltd. All rights reserved. Keywords: Parallel manipulators Instantaneous degrees-of-freedom Singularity Lower-mobility manipulators 1. Introduction Spatial manipulators were originally designed to have all of the possible six-degrees-of-freedom (DoF). The famous PUMA [1] or the Stewart platform manipulators [2] represent this trend. While these manipulators have proven to be versatile in terms of handling a large range of tasks by virtue of their full-mobility, they turned out to be relatively expensive for the same reason. Moreover, in a large number of applications where only a subset of the six-DoF sufficed, the manipulators' capabilities would appear to be practically redundant. It is for these reasons, that a large number of manipulators with “lower-mobility”, i.e., less than six-DoF, have come up in the past few decades: the SCARA [3], the DELTA [4], the 3-R PS [5], the cylindrical manipulator [6], the 3-U PU [7], the Agile Eye [8], the CaPaMan [9] and the MaPaMan [10], to name a few. Most of these manipulators are designed for a particular set of tasks. For example, the Agile Eye, as the name suggests, provides a dexterous camera mount with spherical motion capability; the DELTA, on the other hand, provides three translational motions, making it very successful in operations such as “pick-and-place” in the industries. The motion analysis of these two manipulators, and others with similar motions, is relatively easy, for their workspaces form not only proper subsets of SE(3) (i.e., the special Euclidean group signifying the space of general displacements of the rigid bodies), these are actually sub-groups of the same – SO(3)in the case of Agile Eye, (i.e., the special orthogonal group signifying the space of rigid body rotations) and ℝ 3 for DELTA (i.e., space of pure translations). Understanding the motion of other manipulators, such as the 3-R PS and MaPaMan, is more challenging, for these three-DoF manipulators do not belong to either spherical, or Cartesian categories. Instead, their platforms can show motions which are combinations of rotation(s), as well as translation(s). For instance, the 3-R PS manipulator, even though originally designated as a parallel wrist, actually possesses two rotational, and one translational DoF [11]. The study of the DoF or mobility of linkages and manipulators has a long history. In [12], Gogu has presented a summary of the important developments up to the year 2005. His analysis of a significant number of existing mobility criteria showed that simple Mechanism and Machine Theory 69 (2013) 127–141 ⁎ Corresponding author at: Department of Engineering Design, Indian Institute of Technology Madras, Chennai 600 036, India. Tel.: +91 44 2257 4733. E-mail addresses: rarunsrivatsan@gmail.com (R.A. Srivatsan), sandipan@iitm.ac.in (S. Bandyopadhyay), asitava@mecheng.iisc.ernet.in (A. Ghosal). 0094-114X/$ – see front matter © 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.mechmachtheory.2013.04.016 Contents lists available at ScienceDirect Mechanism and Machine Theory journal homepage: www.elsevier.com/locate/mechmt