Inverse Kinematics Solutions Using Conformal Geometric Algebra

  • Andreas AristidouEmail author
  • Joan Lasenby


This paper describes a novel iterative Inverse Kinematics (IK) solver, FABRIK, that is implemented using Conformal Geometric Algebra (CGA). FABRIK uses a forward and backward iterative approach, finding each joint position via locating a point on a line. We use the IK of a human hand as an example of implementation where a constrained version of FABRIK was employed for pose tracking. The hand is modelled using CGA, taking advantage of CGA’s compact and geometrically intuitive framework and that basic entities in CGA, such as spheres, lines, planes and circles, are simply represented by algebraic objects. This approach can be used in a wide range of computer animation applications and is not limited to the specific problem discussed here. The proposed hand pose tracker is real-time implementable and exploits the advantages of CGA for applications in computer vision, graphics and robotics.


  1. 1.
    Hestens, D., Sobczyk, G.: Clifford Algebra to Geometric Calculus: A Unified Language for Mathematics and Physics. Reidel, Dordrecht (1984) Google Scholar
  2. 2.
    Doran, C., Lasenby, A.: Geometric Algebra for Physicists. Cambridge University Press, Cambridge (2003) zbMATHGoogle Scholar
  3. 3.
    Aristidou, A., Lasenby, J.: Inverse kinematics: a review of existing techniques and introduction of a new fast iterative solver. Cambridge University Department of Engineering Technical Report, CUED/F-INFENG/TR-632 (2009) Google Scholar
  4. 4.
    Dorst, L., Fontijne, D., Mann, S.: Geometric Algebra for Computer Science: An Object-Oriented Approach to Geometry. Morgan Kaufmann, San Mateo (2009) Google Scholar
  5. 5.
    Bayro-Corrochano, E., Kähler, D.: Motor algebra approach for computing the kinematics of robot manipulators. J. Robot. Syst. 17(9), 495–516 (2000) zbMATHCrossRefGoogle Scholar
  6. 6.
    Bayro-Corrochano, E.: Robot perception and action using conformal geometric algebra. In: Handbook of Geometric Computing, pp. 405–458, Chap. 13. Springer, Berlin (2005) CrossRefGoogle Scholar
  7. 7.
    Zamora, J., Bayro-Corrochano, E.: Inverse kinematics, fixation and grasping using conformal geometric algebra. In: Proceedings of the IEEE International Conference on Intelligent Robots and Systems (IROS ’04), vol. 4, pp. 3841–3846 (2004). doi: 10.1109/IROS.2004.1390013 Google Scholar
  8. 8.
    Hildenbrand, D.: Tutorial: Geometric computing in computer graphics using conformal geometric algebra. Comput. Graph. 29(5), 795–803 (2005) CrossRefGoogle Scholar
  9. 9.
    Zamora, J., Bayro-Corrochano, E.: Kinematics and grasping using conformal geometric algebra. In: Lenarčič, J., Roth, B. (eds.) Advances in Robot Kinematics, pp. 473–480. Springer, Berlin (2006) CrossRefGoogle Scholar
  10. 10.
    Hildenbrand, D., Zamora, J., Bayro-Corrochano, E.: Inverse kinematics computation in computer graphics and robotics using conformal geometric algebra. Adv. Appl. Clifford Algebras 18, 699–713 (2008) MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Hildenbrand, D., Fontijne, D., Wang, Y., Alexa, M., Dorst, L.: Competitive runtime performance for inverse kinematics algorithms using conformal geometric algebra. In: Proceedings of Eurographics Conference, 2006 Google Scholar
  12. 12.
    Tanev, T.K.: Geometric algebra approach to singularity of parallel manipulators with limited mobility. In Lenarcic, J., Wenger, P. (eds.) Advances in Robot Kinematics: Analysis and Design, pp. 39–48. Springer, Dordrecht (2008) CrossRefGoogle Scholar
  13. 13.
    Bayro-Corrochano, E., Zamora, J.: Differential and inverse kinematics of robot devices using conformal geometric algebra. Robotica 25(1), 43–61 (2007) CrossRefGoogle Scholar
  14. 14.
    Hildenbrand, D., Lange, H., Stock, F., Koch, A.: Efficient inverse kinematics algorithm based on conformal geometric algebra (using reconfigurable hardware). In: Proceedings of the 3rd International Conference on Computer Graphics Theory and Applications, Madeira, Portugal, 2008 Google Scholar
  15. 15.
    Wang, L.-C.T., Chen, C.C.: A combined optimization method for solving the inverse kinematics problems of mechanical manipulators. IEEE Trans. Robot. Autom. 7(4), 489–499 (1991) CrossRefGoogle Scholar
  16. 16.
    Aristidou, A., Lasenby, J.: FABRIK: a fast, iterative solver for the inverse kinematics problem. Graph. Models 73(5), 243–260 (2011) CrossRefGoogle Scholar
  17. 17.
    PhaseSpace Inc: Optical motion capture systems.
  18. 18.
    Lasenby, A.N., Lasenby, J., Wareham, R.: A covariant approach to geometry using geometric algebra. Cambridge University Department of Engineering Technical Report, CUED/F-INFENG/TR-483 (2004) Google Scholar
  19. 19.
    Lasenby, J., Fitzgerald, W.J., Lasenby, A.N., Doran, C.J.L.: New geometric methods for computer vision: an application to structure and motion estimation. Int. J. Comput. Vis. 26(3), 191–213 (1998) CrossRefGoogle Scholar
  20. 20.
    Aristidou, A.: Tracking and modelling motion for biomechanical analysis. PhD Thesis, University of Cambridge, Cambridge, UK (October 2010) Google Scholar
  21. 21.
    Kaimakis, P., Lasenby, J.: Physiological modelling for improved reliability in silhouette-driven gradient-based hand tracking. In: Proceedings of the International Conference on Computer Vision and Pattern Recognition, Miami, USA, 25 June 2009, pp. 19–26 Google Scholar
  22. 22.
    The Mathworks—MATLAB and Simulink for technical computing.

Copyright information

© Springer-Verlag London Limited 2011

Authors and Affiliations

  1. 1.Department of EngineeringUniversity of CambridgeCambridgeUK

Personalised recommendations