Abstract
In this paper, a new and systematic algorithm is presented for the forward kinematic analysis of parallel mechanisms (PMs) by considering joint motion limitations. The proposed algorithm is based on Euler parameters, the so-called linear implicitization algorithm and interval analysis in seven-dimensional kinematic space. Interval arithmetic computations and solving system of nonlinear interval equations are performed by means of INTLAB and RealPaver software, respectively. The ranges of passive joints motion of PMs are given as input to the algorithm, and feasible solutions of the forward kinematic problem (FKP) are the output of the algorithm. The advantages of combining seven-dimensional kinematic space and interval analysis in the FKP of PM are twofold: eliminating the trigonometric expressions from equations and considering the ranges of passive joints motion. The main contribution of this paper can be regarded as the synergy of seven-dimensional kinematic space and interval analysis which is used to the end of refining the solutions of FKP of PMs by taking into account passive joints limitation. As case studies, the proposed algorithm is implemented and verified, namely the 3-RPR and 4-PUU PMs.
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References
Agrawal SK, Desmier G, Li S (1995) Fabrication and analysis of a novel 3 DOF parallel wrist mechanism. J Mech Des 117:343
Albus J, Bostelman R, Dagalakis N (1993) The NIST robocrane. J Robot Syst 10:709–724
Arai T, Stoughton R, Homma K, Adachi H, Nakamura T, Nakashima K (1991) Development of a parallel link manipulator. In: Fifth international conference on advanced robotics, 1991. Robots in unstructured environments, 91 ICAR. IEEE, pp 839–844
Berti A, Merlet J-P, Carricato M (2013) Solving the direct geometrico-static problem of 3-3 cable-driven parallel robots by interval analysis: preliminary results. In: Cable-driven parallel robots. Springer, pp 251–268
Brandt G et al (1999) CRIGOS: a compact robot for image-guided orthopedic surgery. IEEE Trans Inf Technol Biomed 3:252–260
Brunnthaler K, Schröcker H-P, Husty M (2005) A new method for the synthesis of Bennett mechanisms. In: Proceedings of CK
Chablat D, Wenger P, Majou F, Merlet J-P (2004) An interval analysis based study for the design and the comparison of three-degrees-of-freedom parallel kinematic machines. Int J Robot Res 23:615–624
Craig JJ (2004) Introduction to robotics: mechanics and control. Pearson Prentice Hall, Upper Saddle River
Farzaneh Kaloorazi M, Tale Masouleh M, Caro S (2016) Collision-free workspace of parallel mechanisms based on an interval analysis approach. Robotica 35:1747–1760
Farzaneh Kaloorazi MH, Tale Masouleh M, Caro S (2016) Determining the maximal singularity-free circle or sphere of parallel mechanisms using interval analysis. Robotica 34:135–149
Gan D, Liao Q, Dai JS, Wei S, Seneviratne L (2009) Forward displacement analysis of the general 6–6 Stewart mechanism using Gröbner bases. Mech Mach Theory 44:1640–1647
Gosselin CM, Hamel J-F (1994) The agile eye: a high-performance three-degree-of-freedom camera-orienting device. In: Proceedings 1994 IEEE international conference on robotics and automation, 1994. IEEE, pp 781–786
Gosselin CM, Merlet J-P (1994) The direct kinematics of planar parallel manipulators: special architectures and number of solutions. Mech Mach Theory 29:1083–1097
Gosselin CM, St-Pierre É (1997) Development and experimentation of a fast 3-DOF camera-orienting device. Int J Robot Res 16:619–630
Gosselin C, Kong X, Foucault S, Bonev I (2004) A fully-decoupled 3-dof translational parallel mechanism. Parallel Kinemat Mach Res Pract 24:595–610
Gosselin CM, Tale Masouleh M, Duchaine V, Richard P-L, Foucault S, Kong X (2007) Parallel mechanisms of the multipteron family: kinematic architectures and benchmarking. In: 2007 IEEE international conference on robotics and automation IEEE, pp 555–560
Gouttefarde M, Daney D, Merlet J-P (2011) Interval-analysis-based determination of the wrench-feasible workspace of parallel cable-driven robots. IEEE Trans Robot 27:1–13
Granvilliers L (2002) RealPaver user’s manual, 01st edn. IRIN University of Nantes, Nantes
Granvilliers L, Benhamou F (2006) Algorithm 852: Realpaver: an interval solver using constraint satisfaction techniques. ACM Trans Math Softw (TOMS) 32:138–156
Griffis M, Duffy J (1989) A forward displacement analysis of a class of Stewart platforms. J Robot Syst 6:703–720
Hofschuster W, Krämer WA (1997) Fast public domain interval library in ANSI-C. In: 15th IMACS world congress on scientific computation, modelling and applied mathematics. Berlin, pp 395–400
Huang T, Whitehouse DJ, Wang J (1998) The local dexterity optim architecture and design criteria of parallel machine tools. CIRP Ann Manuf Technol 47:346–350
Husain M, Waldron KJ (1994) Direct position kinematics of the 3-1-1-1 Stewart platforms. J Mech Des 116:1102
Husty ML (1996) An algorithm for solving the direct kinematics of general Stewart-Gough platforms. Mech Mach Theory 31:365–379
Husty ML (2006) Algebraic methods in mechanisms analysis and synthesis. Mach Des Res 31:37–48
Husty ML (2010) Using Geogebra in elementary geometry. In: Sir Z (ed) The 30th Czech conference on geometry and graphics
Husty ML, Schröcker HP (2009) Algebraic geometry and kinematics. In: Emiris I, Sottile F, Theobald T (eds) Nonlinear computational geometry. The IMA volumes in mathematics and its applications, vol 151. Springer, New York, pp 85–107
Husty ML, Schröcker H-P (2013) Kinematics and algebraic geometry. In: 21st century kinematics. Springer, pp 85–123
Husty ML, Pfurner M, Schröcker H-P (2007a) A new and efficient algorithm for the inverse kinematics of a general serial 6R manipulator. Mech Mach Theory 42:66–81
Husty ML, Pfurner M, Schröcker H-P, Brunnthaler K (2007b) Algebraic methods in mechanism analysis and synthesis. Robotica 25:661–675
Innocenti C (1995) Direct kinematics in analytical form of the 6-4 fully-parallel mechanism. J Mech Des 117:89
Innocenti C, Parenti-Castelli V (1993) Direct kinematics in analytical form of a general geometry 5-4 fully-parallel manipulator. In: Angeles J, Hommel G, Kovács P (eds) Computational kinematics. Solid mechanics and its applications, vol 28. Springer, Dordrecht, pp 141–152
Lim K (1997) Forward kinematics solution of Stewart platform using neural networks. Neurocomputing 16:333–349
Kearfott RB, Dawande M, Du K, Hu C (1994) Algorithm 737: INTLIB—a portable Fortran 77 interval standard-function library. ACM Trans Math Softw (TOMS) 20:447–459
Khalilpour SA, Loloei AZ, Taghirad HD, Tale Masouleh M (2013) Feasible kinematic sensitivity in cable robots based on interval analysis. In: Cable-driven parallel robots. Springer, pp 233–249
Knüppel O (1994) PROFIL/BIAS—a fast interval library. Computing 53:277–287
Kong X, Gosselin CM (2005) Type synthesis of 5-DOF parallel manipulators based on screw theory. J Robot Syst 22:535–547
Kong X, Gosselin CM (2007) Type synthesis of parallel mechanisms. Springer, Berlin
Lee T-Y, Shim J-K (2001) Forward kinematics of the general 6–6 Stewart platform using algebraic elimination. Mech Mach Theory 36:1073–1085
Li Y, Xu Q (2005) Design and analysis of a new 3-DOF compliant parallel positioning platform for nanomanipulation. In: 2005 5th IEEE conference on nanotechnology. IEEE, pp 861–864
Lin W, Griffis M, Duffy J (1992) Forward displacement analyses of the 4-4 Stewart platforms. J Mech Des 114:444
Liu X-J, Wang J, Gao F, Wang L-P (2001) On the design of 6-DOF parallel micro-motion manipulators. In: Proceedings 2001 IEEE/RSJ international conference on intelligent robots and systems 2001. IEEE, pp 343–348
Ma LS, Hao KR, Ding YS (2013) Forward kinematics of six-DOF parallel robot based on adaptive differential evolution. Appl Mech Mater 303:1674–1677
Merlet J-P (1996) Direct kinematics of planar parallel manipulators. In: 1996 Proceedings IEEE international conference on robotics and automation, 1996. IEEE, pp 3744–3749
Merlet J-P (2004) Solving the forward kinematics of a Gough-type parallel manipulator with interval analysis. Int J Robot Res 23:221–235
Merlet J-P (2006) Parallel robots. Springer, Berlin
Moore RE, Cloud MJ, Kearfott RB (2009) Introduction to interval analysis. SIAM, Bangkok
Morell A, Tarokh M, Acosta L (2013) Solving the forward kinematics problem in parallel robots using Support Vector Regression. Eng Appl Artif Intell 26:1698–1706
Naderi D, Tale Masouleh M, Varshovi-Jaghargh P (2013) Forward kinematic investigation of three 4-DOF parallel robots with prismatic actuators performing 3T1R motion pattern in seven-dimensional kinematic space (Text in persian). Modares J Mech Eng 13:35–50
Naderi D, Tale Masouleh M, Varshovi-Jaghargh P (2016) Gröbner basis and resultant method for the forward displacement of 3-DoF planar parallel manipulators in seven-dimensional kinematic space. Robotica 34:2610–2628
Nag A, Mohan S, Bandyopadhyay S (2017) Forward kinematic analysis of the 3-RPRS parallel manipulator. In: Wenger P, Flores P (eds) New trends in mechanism and machine science. Mechanisms and machine science, vol 43. Springer, Cham, p 103
Parikh PJ, Lam SS (2005) A hybrid strategy to solve the forward kinematics problem in parallel manipulators. IEEE Trans Robot 21:18–25
Pfurner M (2006) Analysis of spatial serial manipulators using kinematic mapping. Ph. D. Thesis, University Innsbruck
Richard P-L, Gosselin CM, Kong X (2007) Kinematic analysis and prototyping of a partially decoupled 4-DOF 3T1R parallel manipulator. J Mech Des 129:611
Rump SM (1999) INTLAB—interval laboratory. Springer, Berlin
Saafi H, Laribi MA, Arsicault M, Zeghloul S (2015) Forward kinematic model of a new spherical parallel manipulator used as a master device. In: Bai S, Ceccarelli M (eds) Recent advances in mechanism design for robotics. Mechanisms and machine science, vol 33. Springer, Cham, pp 399–406
Siciliano B, Khatib O (2008) Springer handbook of robotics. Springer, Berlin
Stewart D (1965) A platform with six degrees of freedom. Proc Inst Mech Eng 180:371–386
Study E (1891) von den Bewegungen und Umlegungen. Math Ann 39:441–565
Tale Masouleh M (2010) Kinematic analysis of five-DOF (3T2R) parallel mechanisms with identical limb structures. Université Laval
Tale Masouleh M, Husty M, Gosselin C (2010a) Forward kinematic problem of 5-PRUR parallel mechanisms using study parameters. In: Advances in robot kinematics: motion in man and machine. Springer, pp 211–221
Tale Masouleh M, Husty M, Gosselin C (2010b) A general methodology for the forward kinematic problem of symmetrical parallel mechanisms and application to 5-PRUR parallel mechanisms (3T2R). In: ASME 2010 international design engineering technical conferences and computers and information in engineering conference, American Society of Mechanical Engineers, pp 1279–1288
Tale Masouleh M, Gosselin C, Husty M, Walter DR (2011) Forward kinematic problem of 5-RPUR parallel mechanisms (3T2R) with identical limb structures. Mech Mach Theory 46:945–959
Varshovi-Jaghargh P, Naderi D, Tale Masouleh M (2012) Forward kinematic problem of two 4-RRUR with different geometric structures and one 4-RUU spatial parallel robots (Text in Persian). Modares J Mech Eng 12:105–119
Varshovi-Jaghargh P, Naderi D, Tale Masouleh M (2014) Forward kinematic problem of three 4-DOF parallel mechanisms (4-PRUR1, 4-PRUR2 and 4-PUU) with identical limb structures performing 3T1R motion pattern. Sci Iran Trans B, Mech Eng 21:1671–1682
Walster GW (1999) Closed Interval Systems. Sun Microsystems, Santa Clara
Walter DR, Husty ML (2010) On implicitization of kinematic constraint equations. Mach Des Res 26:218–226
Walter DR, Husty ML, Pfurner M (2008) The SNU 3-UPU parallel robot from a theoretical viewpoint. In: Proceedings of the second international workshop on fundamental issues and future research directions for parallel mechanisms and manipulators, pp 1–8
Zhang CD, Song SM (1992) Forward kinematics of a class of parallel (Stewart) platforms with closed-form solutions. J Robot Syst 9:93–112
Zhou W, Chen W, Liu H, Li X (2015) A new forward kinematic algorithm for a general stewart platform. Mech Mach Theory 87:177–190
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Varshovi-Jaghargh, P., Tale Masouleh, M. & Naderi, D. Forward Kinematic Analysis of Parallel Mechanisms in Seven-Dimensional Kinematic Space by Considering Limitation of Passive Joints motion. Iran J Sci Technol Trans Mech Eng 43, 315–329 (2019). https://doi.org/10.1007/s40997-017-0128-7
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DOI: https://doi.org/10.1007/s40997-017-0128-7