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Rigid-Body Motions

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Handbook of Manufacturing Engineering and Technology

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Abstract

Mechanism is made up of links and kinematic pair joints which move in the three-dimensional space. Many physical objects are considered as rigid bodies for the convenience of theoretical analysis. For rigid body motions, the representations of rigid body rotation have a wide range of approaches including the representations from the directional cosine matrix to the exponential coordinates. The kinematics of rigid body is the motion analysis without considering any external forces acting on a rigid body, which is actually the foundation of the dynamics of rigid bodies. This chapter firstly presents the representation method of the position and orientation of rigid bodies using both the algebraic and the geometric methods. Then, an example of a SCARA robot is given to show the applications of the basic theoretical tools for rigid body motions. The purpose of this chapter is to provide the basic mathematical tools and the main results for the kinematics of rigid body though many objects may have elastic deformation in practical engineering problems.

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References

  • Ahlers SG, McCarthy JM (2001) The Clifford algebra and the optimization of robot design. In: Corrochano EB (ed) Geometric algebra with applications in science and engineering, Springer Verlag, Basel, pp 235–251

    Chapter  Google Scholar 

  • Akyar B (2008) Dual quaternions in spatial kinematics in an algebraic sense. Turk J Math 32(4):373–391

    MATH  MathSciNet  Google Scholar 

  • Arribas M, Elipe A, Palacios M (2006) Quaternions and the rotation of a rigid body. Celest Mech Dyn Astron 96(3–4):239–251

    Article  MATH  MathSciNet  Google Scholar 

  • Aspragathos NA, Dimitros JK (1998) A comparative study of three methods for robot kinematics. IEEE Trans Syst Man Cybern 28(2):135–145

    Article  Google Scholar 

  • Ball RS (1998) A treatise on the theory of screws. Cambridge University Press, Cambridge

    MATH  Google Scholar 

  • Brockett RW (1984) Robotic manipulators and the product of exponentials formula. In: Fuhrmann PA (ed) Mathematical theory of networks and systems. Springer, Berlin/Heidelberg

    Google Scholar 

  • Cheng H, Gupta KG (1989) An historical note on finite rotations. J Appl Mech 56:139–145

    Article  MathSciNet  Google Scholar 

  • Clifford WK (1873) Preliminary sketch of bi-quaternions. Proc Lond Math Soc 4:381–395

    MATH  MathSciNet  Google Scholar 

  • Coolidge JL (2003) A history of geometrical methods. Courier Dover, New York

    Google Scholar 

  • Coutsias EA, Romero L (2004) The quaternions with an application to rigid body dynamics. Technical report SAND2004-0153. Sandia National Laboratories

    Google Scholar 

  • Craig JH (2005) Introduction to robotics: mechanics and control. Prentice Hall, New York

    Google Scholar 

  • Dai JS (2006) An historical review of the theoretical development of rigid body displacements from Rodrigues parameters to the finite twist. Mech Mach Theory 41(1):41–52

    Article  MATH  MathSciNet  Google Scholar 

  • Dai JS, Jones J (2001) Interrelationship between screw systems and corresponding reciprocal systems and applications. Mech Mach Theory 36(5):633–651

    Article  MATH  MathSciNet  Google Scholar 

  • Davidson JK, Hunt KKH (2004) Robots and screw theory: applications of kinematics and statics to robotics. Oxford University Press, London

    Google Scholar 

  • Delphenich DH (2012) The representation of physical motions by various types of quaternions. arXiv preprint arXiv:1205.4440

    Google Scholar 

  • Dimentberg FM (1965) The screw calculus and its application to mechanics. US Department of Commerce Translation No AD680993

    Google Scholar 

  • Etzel KR, McCarthy JM (1996) A metric for spatial displacements using biquaternions on SO(4). In: Proceedings of the IEEE international conference on robotics and automation, Minneapolis, 22–28 April

    Google Scholar 

  • Fu KS, Gonzalez RC, Lee CSG (1988) Robotics: control, sensing, vision and intelligence. McGraw-Hill, New York

    Google Scholar 

  • Ge QJ (1998) On the matrix realization of the theory of biquaternions. Trans Am Soc Mech Eng J Mech Des 120:404–407

    Google Scholar 

  • Ge QJ, Varshney A, Menon JP et al (1998) Double quaternions for motion interpolation. In: Proceedings of the ASME design engineering technical conference, Atlanta, Georgia, USA

    Google Scholar 

  • Gouasmi M (2012) Robot Kinematics using Dual Quaternions. IAES Int J Robot Autom 1(1):13–30

    Google Scholar 

  • Hamilton WR II (1844) On quaternions; or on a new system of imaginaries in algebra. Lond Edinb Dublin Philos Mag J Sci 25(163):10–13

    Google Scholar 

  • Hervé JM (1999) The Lie group of rigid body displacements, a fundamental tool for mechanism design. Mech Mach Theory 34(5):719–730

    Article  MATH  MathSciNet  Google Scholar 

  • Huang Z, Li Q, Ding H (2013) Theory of parallel mechanisms. Springer, New York

    Book  Google Scholar 

  • Hunt KH (1990) Kinematic geometry of mechanisms. Clarendon, Oxford

    MATH  Google Scholar 

  • Klein F (1933) Elementary mathematics from an advanced standpoint: arithmetic, algebra, analysis. Bull Am Math Soc 39:495–496

    Article  Google Scholar 

  • Kotelnikov AP (1895) Screw calculus and some applications to geometry and mechanics. Annals of Imperial University of Kazan, Kazan

    Google Scholar 

  • Lie S (1880) Theorie der transformations group I. Math Ann 16(4):441–528

    Article  MATH  MathSciNet  Google Scholar 

  • McCarthy JM (1990) Introduction to theoretical kinematics. MIT Press, New York

    Google Scholar 

  • Mozzi G (1763) Discorso matematico sopra il rotamento momentaneo dei corpi. Donate Campo, Napoli

    Google Scholar 

  • Murray M, Li ZX, Shankar S (1994) A mathematical introduction to robotic manipulation. Taylor & Francis, Boca Raton

    MATH  Google Scholar 

  • O’Reilly OM (2008) Intermediate dynamics for engineers; a unified treatment of Newton-Euler and Lagrangian mechanics. Cambridge University Press, New York/Cambridge

    Book  MATH  Google Scholar 

  • Paul RP, Shimano BE, Mayer G (1981) Kinematic control equations for simple manipulators. IEEE Trans Syst Man Cybern 11:449–455

    Article  Google Scholar 

  • Phillips J (1990) Freedom in machinery: screw theory exemplified, vol 2. Cambridge University Press, London

    Google Scholar 

  • Poinsot L (1851) Théorie nouvelle de la rotation des corps. Bachelier, Paris

    Google Scholar 

  • Rocha CR, Tonetto CP, Dias A (2011) A comparison between the Denavit-Hartenberg and the screw-based methods used in kinematic modeling of robot manipulators. Robot Comp Integr Manuf 27(4):723–728

    Article  Google Scholar 

  • Selig JM (1996) Geometrical methods in robotics. Springer, New York

    Book  MATH  Google Scholar 

  • Senan NAF, O’Reilly OM (2009) On the use of quaternions and Euler-Rodrigues symmetric parameters with moments and moment potentials. Int J Eng Sci 47(4):595–609

    Article  MATH  Google Scholar 

  • Study E (1901) Geometrie der Dynamen. Teubner, Leipzig

    Google Scholar 

  • Tsai LW, Roth B (1973) Incompletely specified displacements: geometry and spatial linkage synthesis. J Eng Ind 95(B):603–611

    Article  Google Scholar 

  • Waldron K, Schmiedeler J (2001) Kinematics. Springer handbook of robotics. Springer, Berlin/Heidelberg

    Google Scholar 

  • Widdows D (2006) Quaternion algebraic geometry. Oxford university, DPhil thesis, Oxford

    Google Scholar 

  • Wu Y, Hu X, Hu D et al (2005) Strapdown inertial navigation system algorithms based on dual quaternions. IEEE Trans Aerosp Electron Syst 41(1):110–132

    Article  MathSciNet  Google Scholar 

  • Yang AT, Freudenstein F (1964) Application of dual-number quaternion algebra to the analysis of spatial mechanisms. J Appl Mech Trans ASME 86(2):300–309

    Article  MathSciNet  Google Scholar 

  • Young L, Duffy J (1986) A theory for the articulation of planar robots. ASME J Mech Transm Autom Des 109(1):29–36

    Article  Google Scholar 

  • Yu JJ, Liu XJ et al (2008) The mathematical foundations of robot mechanism. China Machine Press, Beijing

    Google Scholar 

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Authors and Affiliations

  1. State Key Laboratory of Mechanical System and Vibration, School of Mechanical Engineering, Shanghai Jiao Tong University, 800 Dongchuan Road, 200240, Shanghai, China

    Zhenhua Xiong, Chungang Zhuang & Jianhua Wu

Authors
  1. Zhenhua Xiong
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  2. Chungang Zhuang
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  3. Jianhua Wu
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Corresponding author

Correspondence to Zhenhua Xiong .

Editor information

Editors and Affiliations

  1. Faculty of Engineering, Mechanical Engineering Department, National University of Singapore, Singapore, Singapore

    Andrew Y. C. Nee

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© 2015 Springer-Verlag London

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Xiong, Z., Zhuang, C., Wu, J. (2015). Rigid-Body Motions. In: Nee, A. (eds) Handbook of Manufacturing Engineering and Technology. Springer, London. https://doi.org/10.1007/978-1-4471-4670-4_89

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