IEEE TRANSACTIONS ON ROBOTICS, Vol. 25, No. 4, August 2009, pp. 757-770. A New Assessment of Singularities of Parallel Kinematic Chains Michele Conconi, Marco Carricato Abstract—This paper presents a novel assessment of singu- larities of general parallel kinematic chains. Hierarchical levels in which different critical phenomena originate are recognized. At each level, the causes of singular events are identified and interpreted and, on their basis, a comprehensive taxonomy is proposed. First, the unactuated kinematic chain is studied. The concepts of leg and passive-constraint singularities are described, and stationary and increased-instantaneous-mobility configura- tions are identified. Then, a set of motorized joints is chosen. The effects of first-level singularities on the actuated chain are investigated and further phenomena are identified, such as redundancy singularities and active-constraint singularities. The notions of reaction and action spaces are originally discussed. The consequences on the ability of the actuated chain to effectively govern its local and global freedoms are analyzed, and the complex interactions between the various singular events are studied. The instantaneous redundancy of actuators, occurring when these work either against each other or against joint constraints, is also evaluated. Finally, when the input/output mechanism is considered, the events described at the previous stages are interpreted within the perspective of the machine’s desired use. Index Terms—Kinematic Singularities; Parallel Robots; Robot Kinematics; Taxonomy. I. I NTRODUCTION S INGULARITIES are critical configurations in which the kinetostatic behavior of a mechanism suddenly changes with respect to a full-cycle condition. There is a vast literature dedicated to the subject, which has been studied under several perspectives. Only the most relevant contributions for the purposes of this study, however, are recalled hereafter. For more detailed bibliographic records and further discussions, the reader may refer, for instance, to [1] and [2]. A. Singularities of the input-output kinematic map The singularities exhibited by a serial chain, occurring when the rank of the Jacobian matrix mapping the joint rates to the end-effector twist drops below its maximum possi- ble value [3]–[5], may be readily identified and interpreted. The singularities of a closed-chain mechanism pose, instead, substantially more complicated problems. One of the first attempts to provide a general framework for their study and classification may be traced back to Gosselin and Angeles [6], who derive the input-output velocity map for a generic mechanism by differentiating the implicit equation relating M. Conconi and M. Carricato are with DIEM - Dept. of Mechanical En- gineering, University of Bologna, Italy; website: www.diem.ing.unibo.it/grab; e-mail: {michele.conconi, marco.carricato}@unibo.it. An early version of this paper was presented at 11th Int. Symp. on Advances in Robot Kinematics, June 22-26, 2008, Batz-sur-Mer, France. the input and output configuration variables. In this way, distinct Jacobian matrices are obtained for the inverse and the direct first-order kinematics and different roles played by the corresponding singularities are clearly shown. Indeed, while inverse singularities parallel the degenerate configurations of a serial arm, being naturally associated with the local loss of dexterity of the output link, the direct ones are peculiar to closed-loop chains and are related to the disruptive loss of control of the end-effector. Further insight into such phenomena is given by Kumar [7], who, drawing on the studies by Mohamed and Duffy [8] and Lipkin and Duffy [9], provides a particularly effective screw-theory-based analysis of the instantaneous kinematics of parallel mechanisms. Though describing the same phenomena pointed out by Gosselin and Angeles, Kumar is able to better interpret them, intrinsically relating the mathematical description of the kinetostatic behavior of the mechanism to the physical causes that are at the basis of it. Screw vectors provide, indeed, a most natural and compact way to represent the velocity states of rigid bodies and the systems of forces acting on them. By screw theory, kinematic singularities are associated with the degeneracy of screw subspaces, which may be efficiently studied by the tools of screw geometry [10], [11] or its special instance that is line geometry [12]–[14] 1 . B. Singularities of the overall kinematic map The intrinsic drawback of all approaches basing the singu- larity study on the input-output velocity map consists in that, while focusing on the parameters that are the most relevant for control purposes, they are not able to detect critical phenomena related to other variables, namely those joint parameters that, in closed-loop chains, are neither actuated nor considered as output freedoms (the so-called passive freedoms). Zlatanov, Fenton and Benhabib [17]–[19] clearly show the necessity to set the singularity study in a more general framework, involving the first-order kinematics of all joints present in the mechanism. Instead of resorting to methods that automatically eliminate the passive variables to provide plain input-output relationships, they study the entire system obtained by differentiating the loop-closure equations of the mechanism, identifying any configuration causing the rank reduction of such a system as kinematically singular. Since the first-order derivatives of all variables within the mechanism are considered, the study is necessarily exhaustive. On the other 1 It is worth observing that in [14] direct kinematic singularities are erroneously addressed as uncertainty configurations and the same confusion may be found elsewhere in the literature (for instance, in [15] and [16]). However, Hunt [11] refers to an altogether different concept when he first introduces the latter locution, as detailed in the following.