Geometry and kinematics for a spherical-base integrated parallel mechanism

Meccanica (2016) 51:1607–1621 DOI 10.1007/s11012-016-0438-7 RECENT PROGRESS AND NOVEL APPLICATIONS OF PARALLEL MECHANISMS Geometry and kinematics for a spherical-base integrated parallel mechanism Jie Sun . Xinsheng Zhang . Guowu Wei . Jian S. Dai Received: 27 October 2014 / Accepted: 2 March 2016 / Published online: 4 May 2016 Ó The Author(s) 2016. This article is published with open access at Springerlink.com Abstract Parallel mechanisms, in general, have a rigid base and a moving platform connected by several limbs. For achieving higher mobility and dexterity, more degrees of freedom are introduced to the limbs. However, very few researchers focus on changing the design of the rigid base and making it foldable and reconfigurable to improve the performance of the mechanism. Inspired by manipulating an object with a metamorphic robotic hand, this paper presents for the first time a parallel mechanism with a reconfigurable base. This novel spherical-base integrated parallel mechanism has an enlarged workspace compared with traditional parallel manipulators. Evolution and structure of the proposed parallel mechanism is introduced and the geometric constraint of the mechanism is investigated based on mechanism decomposition. Further, kinematics of the proposed mechanism is reduced to the solution of a univariate polynomial of degree 8. Moreover, screw theory based Jacobian is presented followed by the velocity analysis of the mechanism. J. Sun  X. Zhang  J. S. Dai (&) Centre for Robotics Research, King’s College London, University of London, Strand, London WC2R 2LS, UK e-mail: G. Wei School of Computing, Science and Engineering, University of Salford, Salford, Manchester M5 4WT, UK Keywords Parallel mechanism  Reconfigurable base  Metamorphic hand  Kinematics  Screw theory  Jacobian 1 Introduction A typical parallel mechanism consists of a moving platform that is connected to a fixed base by several (at least two) limbs or legs. In general, the moving platform of parallel mechanisms has both rotational and translational motion [1, 2]. However, in order to reduce the complexity and cater some specific applications, the low-mobility parallel mechanisms [3–6] have drawn numerous interests from researchers in mechanism and machine design. In particular, Chablat and Wenger [7] proposed a 3-DOF parallel mechanism to realise three axes rapid machining applications. Zhao et al. [8, 9] investigated three and four DOFs parallel mechanisms relying on equivalent screw groups. Kong and Gosselin [10] presented several parallel mechanisms relying on screw theory based type synthesis method. Similarly, Xu and Li [11] applied screw theory to analyse the mobility and stiffness of an over-constrained 3-PRC parallel mechanism and converted it into a non-over-constrained 3-CTC parallel mechanism of the same mobility and kinematic properties. Huda and Takeda [12] invented a 3-URU parallel mechanism with three dimensional rotations. Such parallel mechanisms were widely adopted to achieve wrist-like motion, such as Argos, 123 1608 proposed by Vischer and Clavel [13] and the 3-RUU mechanism, proposed by Gregorio [14]. Gan and Dai [15] studied constraint screw systems of a 3-PUP parallel mechanism and revealed the influence between them and limb arrangements. Zhang et al. [16] discussed the constraint singularity and analysed the bifurcated motion of a 3-PUP parallel mechanism and the conversion between two bifurcated motion branches. In addition, some redundant parallel mechanisms [17, 18] were put forward to avoid singularities and obtain better kinematic properties. The parallel mechanism mentioned above are all composed of rigid base and non-reconfigurable moving platform. In other words, their base or moving platform is a component with zero DOF rather than a mechanism with additional moving capability. Recently, the parallel mechanisms with reconfigurable features have been capturing attentions from the researchers in the fields of mechanisms and robotics. Based on the concept reconfigurability and principle of metamorphosis [19], Gan et al. [20] proposed a reconfigurable Hooke (rT) joint and presented a new metamorphic parallel mechanism that was capable of changing mobility and topological configurations. Zhang et al. [21] identified an axis-variable (vA) joint based on origami fold [22] leading to the development of a metamorphic parallel mechanism that had the capability of changing its mobility from 3 to 6 DOF. Wei and Dai [23] proposed a variable revolute (vR) joint with application to the constructure of a family of reconfigurable and deployable Platonic mechsnisms. In addition, there is another kind of metamorphic parallel mechanisms that can reconfigure themselves through changing the configurations of their moving platform. Yi et al. [24] presented a flexible folded parallel gripper to meet the requests of both grasping and positioning objects with irregular shape and size. Mohamed and Gosselin [25] presented a kind of parallel mechanisms with reconfigurable platforms and analysed redundancy of proposed parallel mechanisms. Lambert [26] presented and analysed mobility and kinematics of a PentaG robot, which is a parallel mechanism with a flexible planar platform providing 5 DOFs in total. In contrast to the above flexible-platform parallel mechanisms, the concept of parallel mechanisms with a foldable/reconfigurable base can be brought up but no literature shows the relevant investigation. Inspired by the grasp and manipulation of an object with a metamorphic hand containing a reconfigurable palm 123 Meccanica (2016) 51:1607–1621 (Fig. 1) [27–31], in this paper, a parallel mechanism with a reconfigurable base is for the first time proposed. The base of this parallel mechanism is formed by a spherical five-bar linkage, which provides augmented motion for each limb. Structure design of the proposed spherical-base integrated parallel mechanism is introduced, and geometry and kinematics of the mechanism are investigated leading to closed-form solutions. Screw theory [32] based Jacobian is then presented followed by the velocity analysis. 2 A spherical-base integrated parallel mechanism 2.1 From manipulation with a metamorphic hand to a parallel mechanism with a reconfigurable base Figure 1 illustrates a three-fingered metamorphic robotic hand grasping and manipulating a disk. The metamorphic robotic hand (Metahand) contains a reconfigurable palm and three two-phalanx fingers. The reconfigurable palm is formed by a spherical fivebar linkage, with link AE as a grounded link, and the other four links are symmetrically arranged with respect to link AE such that links AB and ED are of the same length and so do links BC and DC. The three fingers are respectively mounted on link AE at point A1, on link DC at point A2 and on link BC at point A3. When the palm is in a configuration that all the links are in a same plane, the three points A1, A2 and A3 are Fig. 1 Object (...truncated)