Computational algebraic geometry and global analysis of regional manipulators Teijo Arponen a , Andreas Müller b , Samuli Piipponen a,⇑ , Jukka Tuomela a a Department of Physics and Mathematics, University of Eastern Finland, PO Box 111, 80101 Joensuu, Finland b University of Michigan-Shanghai Jiao Tong University Joint Institute, 800 Dong Chuan Road, Shanghai 200240, PR China article info Keywords: Regional manipulators Kinematical analysis Algebraic geometry Gröbner bases Ideal decomposition abstract The global analysis of the singularities of regional manipulators is addressed in this paper. The problem is approached from the point of view of computational algebraic geometry. The main novelty is to compute the syzygy module of the differential of the constraint map. Composing this with the differential of the forward kinematic map and studying the associated Fitting ideals allows for a complete stratification of the configuration space according to the corank of singularities. Moreover using this idea we can also compute the boundary of the image of the forward kinematic map. Obviously this gives us also a description of the image itself, i.e. the manipulator workspace. The approach is feasible in practice because generators of syzygy modules can be computed in a similar way as Gröbner bases of ideals. Ó 2014 Elsevier Inc. All rights reserved. 1. Introduction Robot manipulators inevitably possess forward kinematic singularities. It is therefore desirable to ensure at least that a manipulator does not possess higher-order forward kinematic singularities. Since this is known to be a generic property [1], such manipulators are called generic. That is, typically their set of forward kinematic singularities is a smooth manifold. Most industrial manipulators are non-generic, however, in the sense that their singular variety is not a smooth manifold. The anal- ysis of kinematic singularities of serial manipulators is commonly attempted by means of local approaches [2,3]. The strat- ification of the singular set according to the corank of singularities of regional manipulators was introduced by Pai and Leu [1]. Global analysis was only presented for particular regional manipulators [4–6]. The kinematics of a closed loop mechanism is described by a system of constraint equations that define its configuration space (c-space) – the variety generated by the constraints. This c-space is in general not a smooth manifold but possesses singular points, which correspond to points where the mechanism’s instantaneous mobility changes. Hence the singularities of the c-space variety are critical configurations that interfere with the operation of the mechanism. The kinematics of a se- rial manipulator (i.e. a serial kinematic chain) is described by its forward kinematics mapping. Its configuration space is a priori a smooth manifold. Serial manipulators exhibit forward kinematic singularities, which are merely the critical points of the forward kinematic mapping forming a subvariety R within the c-space. From an application point of view it is imper- ative to characterize the singular variety in regard to its manifold structure. In particular if R is a smooth manifold, then the manipulator’s motion is well-defined when passing through a singularity. Since the singular variety is also defined by the http://dx.doi.org/10.1016/j.amc.2014.01.138 0096-3003/Ó 2014 Elsevier Inc. All rights reserved. ⇑ Corresponding author. E-mail addresses: teijo.arponen@uef.fi (T. Arponen), andreas.muller@ieee.org (A. Müller), samuli.piipponen@uef.fi (S. Piipponen), jukka.tuomela@uef.fi (J. Tuomela). Applied Mathematics and Computation 232 (2014) 820–835 Contents lists available at ScienceDirect Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc