Abstract
A method for the kinematic analysis of parallel manipulators, based on Denavit-Hartenberg formalism, is proposed. This formulation allows to model separately each joint-link train (defined as the consecutive structural elements starting from the origin of the reference frame associated with the base, to the origin of the reference frame of the mobile platform through one of the links). This methodology has the main advantage of determining all the generalized variables characterizing the model (in order to evaluate the mechanical limits of the passive joints, their characteristic parameters), and not only those associated with the active joints. In this way, it is possible to model manufacturing and assembly errors in the mechanical components, kinematics errors in the actuators.
Several parallel machinesm according to the adopted classification, have been modeled mapping the stiffness index above the workspace.
The workspace is defined as the Boolean function w: ℜ6 → {0,1} of all the reachable points (x, y, z, ψ, θ, ϕ) considering physical constraints on joints and legs. The set of reachable poses defines the workspace as a connected set of points. The stiffness index was computed considering the scalar value given by the inverse of the condition number k of the Jacobian matrix. This number gives the ratio between the maximum value of the stiffness computed in the direction of the maximum stiffness and the minimum value of stiffness computed in the direction of the minimum stiffness for each point of the closed set of points belonging to the workspace. When 1/k is close to zero it means either that the machine is near a singular configuration or that the stiffness significantly varies along different directions. In both cases, and supposing the manipulator is used for machining operations (i.e.: as a milling machine), the further is 1/k from the unit the worse is the behavior of the machine. This mapping so can be used for a rough evaluation of the usable workspace.
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References
Stoughton R, Arai T 1993 A Modified Stewart Platform Manipulator with Improved Dexterity. IEEE Transactions on Robotics and Automation 9: 166 - 172
Bhattacharya S, Hatwal H, Ghosh A 1995 On the Optimum Design of Stewart Platform Type Parallel Manipulators. Robotica 13:133: 140
Huang T, Whitehouse T J, Wang J 1998 The Local Dexterity. Optimal Architecture and Design Criteria of Parallel Machine Tools. Annals of the CIRP 47: 347 - 351
Merlet J-P 1990 Les Robots parallèles, Hermès, Paris
Fichter EF 1986 A Stewart Platform-Based Manipulator: General Theory and Practical Construction. The International Journal of Robotics Research 5:157- 182
Dafaoui El-M, Amirat Y, Pontnau J, Francois C 1998 Analysis and Design of a Six-DOF Parallel Manipulator, Modeling, Singular Configurations, and Workspace. IEEE Transactions on Robotics and Automation 14: 78 - 91
Dasgupta B, Mruthyunjaya T S 1996 A Constructive Predictor-Corrector Algorithm for the Direct Position Kinematic Problem for a General 6-6 Stewart Platform. Mech. Mack Theory 31: 799 - 811
Nanua P, Waldron K J, Murthy V 1990 Direct Kinematic Solution of a Stewart Platform. IEEE Transactions on Robotics and Automation 6: 438 - 443
Gosselin C 1990 Stiffness Mapping for Parallel manipulators. IEEE Transactions on Robotics and Automation 6: 377 - 382
Merlet J-P 1995 Determination of the Orientation Workspace of Parallel Manipulators. Journal of Intelligent and Robotic Systems 13: 143 - 160
Wang Li-Chun T, Hsieh J 1998 Extreme Reaches and Reachable Workspace Analysis of General Parallel Robotic Manipulators. Journal of Robotic Systems 15 (3): 145 - 159
Conti JP, Clinton CM, Zhang G, Wavering AJ 1997 Dynamic Variation of the Workspace of an Octahedral Hexapod Machine Tool During Machining. Technical Research Report, T.R. 97-28, ISR
Faugere J C, Lazard D 1995 Combinatorial Classes of Parallel Manipulators. Meck Mach. Theory 30: 765 - 776
Innocenti C, Parenti Castelli V 1994 Exhaustive Enumeration of Fully Parallel Kinematic Chains. Dynamic Systems and Control 55: 1135 - 1141
Patel A J, Ehmann K F 1997 Volumetric Error Analysis of a Stewart Platform- Based Machine Tool. Annals of the CIRP 46: 287 - 290
Wang J, Masory O 1993 On the Accuracy of a Stewart Platform - Part I: The Effect of Manufacturing Tolerances. In: Proceedings 1993 IEEE International Conference on Robotics and Automation, Vol I, pp 114 - 120
Masory O, Wang J, Zhuang H 1993 On the Accuracy of a Stewart Platform - Part II: Kinematic Calibration and Compensation. In: Proceedings 1993 IEEE International Conference on Robotics and Automation, Vol I, pp 725 - 731
Molinari Tosatti L, Bianchi G, Fassi I, Boer C R, Jovane F 1998 An Integrated Methodology for the Design of Parallel Kinematic Machines (PKM). Annals of the CIRP 47: 341 - 345
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© 1999 Springer-Verlag London Limited
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Negri, S., Di Bernardo, G., Fassi, I., Molinari Tosatti, L., Bianchi, G., Boër, C.R. (1999). Kinematic Analysis of Parallel Manipulators. In: Boër, C.R., Molinari-Tosatti, L., Smith, K.S. (eds) Parallel Kinematic Machines. Advanced Manufacturing. Springer, London. https://doi.org/10.1007/978-1-4471-0885-6_5
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DOI: https://doi.org/10.1007/978-1-4471-0885-6_5
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