Skip to main content
SpringerLink
Log in

Universally manipulable body models—dual quaternion representations in layered and dynamic MMCs

  • Published:
Autonomous Robots Aims and scope Submit manuscript

Abstract

Surprisingly complex tasks can be solved using a behaviour-based, reactive control system, i.e., a system that operates without an explicit internal representation of the environment and the own body. Nevertheless, application of internal representations has gained interest in recent years because such internal representations can be used to solve problems of perception and motor control (sensor fusion, inverse modeling) and may in addition be applied to higher cognitive functions as are the ability to plan ahead. To endow such a system with the ability to find new behavioural solutions to a given problem in a broad range of possibilities, the internal representation must be universally manipulable, i.e. the model should be able to simulate all movements that are physically possible for the body given. Using recurrent neural networks, models showing this faculty have been proposed being based on the principle of mean of multiple computation (MMC). The extension of this approach to three dimensions requires the introduction of a joint angle representation which allows for computation of mean values. Here we use dual quaternions that are singularity-free and unambiguous which allow for shortest path interpolation. In addition, it has been shown that dual quaternions are the most efficient and most compact form for representing rigid transformations. The model can easily be adapted to bodies of arbitrary geometries. The extended MMC net introduced in this article represents a holistic system that can—following the principle of pattern completion—likewise be used as an inverse model, a forward model, for sensor fusion or other, related capabilities.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • Acosta-Calderon, C., & Hu, H. (2005). Robot imitation: Body schema and body percept. Journal of Applied Bionics and Biomechanics, 2(3–4), 131–148.

    Article  Google Scholar 

  • Aspragathos, N. A., & Dimitros, J. K. (1998). A comparative study of three methods for robot kinematics. IEEE Transactions on Systems, Man, and Cybernetics, Part B, 28(2), 135–145.

    Article  Google Scholar 

  • Atkeson, C. G., & Hollerbach, J. M. (1985). Kinematic features of unrestrained vertical arm movements. Journal of Neurosciences, 5(9), 2318–2330.

    Google Scholar 

  • Bernstein, N. A. (1967). The co-ordination and regulation of movements. Oxford: Pergamon Press Ltd.

    Google Scholar 

  • Biess, A., Liebermann, D. G., & Flash, T. (2007). A computational model for redundant human three-dimensional pointing movements: Integration of independent spatial and temporal motor plans simplifies movement dynamics. Journal of Neurosciences, 27(48), 13,045–13,064.

    Google Scholar 

  • Blanke, O., Landis, T., Spinelli, L., & Seeck, M. (2004). Out-of-body experience and autoscopy of neurological origin. Brain, 127(2), 243–258.

    Article  Google Scholar 

  • Blanke, O., Mohr, C., Michel, C. M., Pascual-Leone, A., Brugger, P., Seeck, M., Landis, T., & Thut, G. (2005). Linking out-of-body experience and self processing to mental own-body imagery at the temporoparietal junction. Journal of Neurosciences, 25(3), 550–557.

    Google Scholar 

  • Bläsing, B. (2006). Crossing large gaps: A simulation study of stick insect behaviour. Adaptive Behaviour, 14(3), 265–285.

    Article  Google Scholar 

  • Bläsing, B., & Cruse, H. (2004). Stick insect locomotion in a complex environment: Climbing over large gaps. The Journal of Experimental Biology, 207, 1273–1286.

    Article  Google Scholar 

  • Blohm, G., & Crawford, J. D. (2007). Computations for geometrically accurate visually guided reaching in 3-D space. Journal of Vision, 7(5), 1–22.

    Article  Google Scholar 

  • Bockemühl, T., Troje, N., & Dürr, V. (2010). Inter-joint coupling and joint angle synergies of human catching movements. Human Movement Science, 29(1), 73–93.

    Article  Google Scholar 

  • Bottema, O., & Roth, B. (1979). Theoretical kinematics. Amsterdam: North-Holland.

    MATH  Google Scholar 

  • Botvinick, M., & Cohen, J. (1998). Rubber hands ‘feel’ touch that eyes see. Nature, 391(6669), 756–756.

    Article  Google Scholar 

  • Brooks, R. A. (1991a). Intelligence without reason. In J. Myopoulos & R. Reiter (Eds.), Proceedings of the 12th international joint conference on artificial intelligence (IJCAI-91) (pp. 569–595). San Mateo: Morgan Kaufmann.

    Google Scholar 

  • Brooks, R. A. (1991b). Intelligence without representation. Artificial Intelligence, 47, 139–159.

    Article  Google Scholar 

  • Chasles, M. (1830). Note sur les propriétés générales du système de deux corps semblables entr’eux et placés d’une manière quelconque dans l’espace; et sur le déplacement fini ou infiniment petit d’un corps solide libre. Bulletin des Sciences Mathematiques, Astronomiques, Physiques et Chimiques, 14(321–326).

  • Clifford, W. (1882). Mathematical papers. London: Macmillan.

    Google Scholar 

  • Cothros, N., Wong, J. D., & Gribble, P. L. (2006). Are there distinct neural representations of object and limb dynamics? Experimental Brain Research, 173(4), 689–697.

    Article  Google Scholar 

  • Cruse, H. (1979). The control of the anterior extreme position of the hindleg of a walking insect. Physiological Entomology, 4, 121–124.

    Article  Google Scholar 

  • Cruse, H. (1986). Constraints for joint angle control of the human arm. Biological Cybernetics, 54, 125–132.

    Article  Google Scholar 

  • Cruse, H. (1999). Feeling our body—the basis of cognition? Evolution and Cognition, 5(2), 162–173.

    Google Scholar 

  • Cruse, H. (2003). The evolution of cognition: A hypothesis. Cognitive Science, 27(1), 135–155.

    MathSciNet  Google Scholar 

  • Cruse, H., & Brüwer, M. (1987). The human arm as a redundant manipulator: The control of path and joint angles. Biological Cybernetics, 57(1–2), 137–144.

    Article  Google Scholar 

  • Cruse, H., & Hübner, D. (2008). Selforganizing memory: Active learning of landmarks used for navigation. Biological Cybernetics, 99(3), 219–236.

    Article  MATH  Google Scholar 

  • Cruse, H., & Steinkühler, U. (1993). Solution of the direct and inverse kinematic problems by a common algorithm based on the mean of multiple computations. Biological Cybernetics, 69, 345–351.

    Article  Google Scholar 

  • Daniilidis, K. (1999). Hand-eye calibration using dual quaternions. International Journal of Robotics Research, 18, 286–298.

    Article  Google Scholar 

  • Davidson, P. R., & Wolpert, D. M. (2004). Internal models underlying grasp can be additively combined. Experimental Brain Research, 155(3), 334–340.

    Article  Google Scholar 

  • de Vignemont, F. (2010). Body schema and body image–pros and cons. Neuropsychologia, 48(3), 669–680.

    Article  Google Scholar 

  • Desmurget, M., & Grafton, S. (2000). Forward modeling allows feedback control for fast reaching movements. Trends in Cognitive Sciences, 4(11), 423–431.

    Article  Google Scholar 

  • Flash, T., & Hogan, N. (1985). The coordination of arm movements: An experimentally confirmed mathematical model. Journal of Neurosciences, 5(7), 1688–1703.

    Google Scholar 

  • Foley, J. D., van Dam, A., Feiner, S. K., & Hughes, J. F. (1996). Computer graphics: Principles and practice in C (2nd ed.). Upper Saddle River: Pearson.

    MATH  Google Scholar 

  • Frith, C. D., Blakemore, S. J., & Wolpert, D. M. (2000). Abnormalities in the awareness and control of action. Philosophical Transactions of the Royal Society of London: Biological Sciences, 355, 1771–1788.

    Article  Google Scholar 

  • Funda, J., & Paul, R. (1990). A computational analysis of screw transformations in robotics. IEEE Transactions on Robotics and Automation, 6(3), 348–356.

    Article  Google Scholar 

  • Ghahramani, Z., & Wolpert, D. M. (1997). Modular decomposition in visuomotor learning. Nature, 386(6623), 392–395.

    Article  Google Scholar 

  • Glenberg, A. M. (1997). What memory is for. Behavioural and Brain Sciences, 20(1).

  • Govindu, V. M. (2004). Lie-algebraic averaging for globally consistent motion estimation. In Computer vision and pattern recognition, IEEE computer society conference on (Vol. 1, pp. 684–691).

    Google Scholar 

  • Grush, R. (2004). The emulation theory of representation: Motor control, imagery, and perception. Behavioural and Brain Sciences, 27, 377–442.

    Google Scholar 

  • Hamilton, W. (1844). On quaternions. In Proceedings of the Royal Irish Academy.

    Google Scholar 

  • Hamilton, W. (1866). Elements of quaternions. London: Longmans Green. New York: Chelsea, 1969.

    Google Scholar 

  • Hanson, A. J. (2005). The Morgan Kaufmann series in interactive 3D technology, Visualizing quaternions. San Mateo: Morgan Kaufmann.

    Google Scholar 

  • Harnad, S. (1990). The symbol grounding problem. Physica D, 42, 335–346.

    Article  Google Scholar 

  • Hartmann, G., & Wehner, R. (1995). The ant’s path integration system: A neural architecture. Biological Cybernetics, 73(6), 483–497.

    MATH  Google Scholar 

  • Hesslow, G. (2002). Conscious thought as simulation of behaviour and perception. Trends in Cognitive Sciences, 6(6), 242–247.

    Article  Google Scholar 

  • Hoffmann, M., Marques, H., Arieta, A. H., Sumioka, H., Lungarella, M., & Pfeifer, R. (2010). Body schema in robotics: A review. IEEE Transactions on Autonomous Mental Development, 2(4), 304–324.

    Article  Google Scholar 

  • Honegger, H. W. (1981). A preliminary note on a new optomotor response in crickets: Antennal tracking of moving targets. Journal of Comparative Physiology A: Neuroethology, Sensory, Neural, and Behavioural Physiology, 142(3), 419–421.

    Article  Google Scholar 

  • Imamizu, H., & Kawato, M. (2008). Neural correlates of predictive and postdictive switching mechanisms for internal models. Journal of Neurosciences, 28(42), 10,751–10,765.

    Google Scholar 

  • Jeannerod, M. (1999). To act or not to act: Perspectives on the representation of actions. Quarterly Journal of Experimental Psychology, 52A, 1–29.

    Article  Google Scholar 

  • Kavan, L., & Žára, J. (2005). Spherical blend skinning: A real-time deformation of articulated models. In I3D ’05: Proceedings of the 2005 symposium on interactive 3D graphics and games (pp. 9–16). New York: ACM.

    Chapter  Google Scholar 

  • Kavan, L., Collins, S., O’Sullivan, C., & Žára, J. (2006). Dual quaternions for rigid transformation blending (Technical report TCD-CS-2006-46). Trinity College Dublin.

  • Kavan, L., Collins, S., Žára, J., & O’Sullivan, C. (2007). Skinning with dual quaternions. In 2007 ACM SIGGRAPH symposium on interactive 3D graphics and games (pp. 39–46). New York: ACM Press.

    Google Scholar 

  • Kavan, L., Collins, S., Žára, J., & O’Sullivan, C. (2008). Geometric skinning with approximate dual quaternion blending. ACM Transactions on Graphics, 27(4), 105.

    Article  Google Scholar 

  • Kawato, M. (1999). Internal models for motor control and trajectory planning. Current Opinion in Neurobiology, 9, 718–727.

    Article  Google Scholar 

  • Kawato, M., & Gomi, H. (1992). The cerebellum and VOR/OKR learning models. Trends in Neurosciences, 15(11), 445–453.

    Article  Google Scholar 

  • Kindermann, T., & Cruse, H. (2002). MMC—a new numerical approach to the kinematics of complex manipulators. Mechanism and Machine Theory, 37(4), 375–394.

    Article  MATH  Google Scholar 

  • Klein Breteler, M., & Meulenbroek, R. (2006). Modeling 3D object manipulation: Synchronous single-axis joint rotations? Experimental Brain Research, 168(3), 395–409.

    Article  Google Scholar 

  • Krakauer, J. W., Ghilardi, M. F., & Ghez, C. (1999). Independent learning of internal models for kinematic and dynamic control of reaching. Nature Neuroscience, 2(11), 1026–1031.

    Article  Google Scholar 

  • Luenberger, D. G. (1984). Linear and nonlinear programming. Reading: Addison-Wesley.

    MATH  Google Scholar 

  • Makarov, V., Song, Y., Velarde, M., Hübner, D., & Cruse, H. (2008). Elements for a general memory structure: Properties of recurrent neural networks used to form situation models. Biological Cybernetics, 98(5), 371–395.

    Article  MATH  MathSciNet  Google Scholar 

  • Makin, T. R., Holmes, N. P., & Ehrsson, H. H. (2008). On the other hand: Dummy hands and peripersonal space. Behavioural Brain Research, 191(1), 1–10.

    Article  Google Scholar 

  • Mataric, M. J. (1999). Behaviour-based robotics. In R. A. Wilson & F. C. Keil (Eds.), MIT encyclopedia of cognitive sciences (pp. 74–77). Cambridge: MIT Press.

    Google Scholar 

  • Mataric, M. J. (2002). Situated robotics. In Encyclopedia of cognitive science. London: Nature Publishing Group, Macmillan Reference Limited.

    Google Scholar 

  • Matheson, T., & Dürr, V. (2003). Load compensation in targeted limb movements of an insect. Journal of Experimental Biology, 206, 3175–3186.

    Article  Google Scholar 

  • Maxwell, E. A. (1951). General homogeneous coordinates in space of three dimensions. Cambridge: Cambridge University Press.

    MATH  Google Scholar 

  • McCarthy, J. (1990). Introduction to theoretical kinematics. Cambridge: MIT Press.

    Google Scholar 

  • McFarland, D., & Bösser, T. (1993). Intelligent behaviour in animals and robots. Cambridge: MIT Press.

    Google Scholar 

  • Metzinger, T. (2006). Different conceptions of embodiment. Psyche, 12(4).

  • Miall, R., Weir, D., Wolpert, D., & Stein, J. (1993). Is the cerebellum a Smith predictor? Journal of Motor Behaviour, 25(3), 203–216.

    Article  Google Scholar 

  • Morasso, P. (1981). Spatial control of arm movements. Experimental Brain Research, 42(2), 223–227.

    Article  Google Scholar 

  • Morasso, P., & Sanguineti, V. (1994). Self-organizing topographic maps and motor planning. In SAB94: Proceedings of the third international conference on simulation of adaptive behaviour: from animals to animats (Vol. 3, pp. 214–220). Cambridge: MIT Press.

    Google Scholar 

  • Muller, C. M. P., Brenner, E., & Smeets, J. B. J. (2009). Maybe they are all circles: Clues and cues. Journal of Vision, 9(9), 10.1-5. doi:10.1167/9.9.10.

    Article  Google Scholar 

  • Murray, R. M., Li, Z., & Sastry, S. S. (1994). A mathematical introduction to robotic manipulation. Boca Raton: CRC.

    MATH  Google Scholar 

  • Mussa-Ivaldi, F., Morasso, P., & Zaccaria, R. (1988). Kinematic networks distributed model for representing and regularizing motor redundancy. Biological Cybernetics, 60(1), 1–16.

    Google Scholar 

  • Niven, J. E., Buckingham, C. J., Lumley, S., Cuttle, M. F., & Laughlin, S. B. (2009). Visual targeting of forelimbs in ladder-walking locusts. Current Biology, 20(1), 86–91.

    Article  Google Scholar 

  • Page, K. L., Zakotnik, J., Durr, V., & Matheson, T. (2008). Motor control of aimed limb movements in an insect. Journal of Neurophysiology, 99(2), 484–499.

    Article  Google Scholar 

  • Pfeifer, R., & Scheier, C. (2001). Understanding intelligence. Cambridge: MIT Press.

    Google Scholar 

  • Rosenbaum, D. A., Engelbrecht, S., Bushe, M., & Loukopoulos, L. (1993). Knowledge model for selecting and producing reaching movements. Journal of Motor Behaviour, 25, 217–227.

    Article  Google Scholar 

  • Rosenbaum, D. A., Loukopoulos, L., Meulenbroek, R., Vaughan, J., & Engelbrecht, S. (1995). Planning reaches by evaluating stored postures. Psychological Review, 102, 28–67.

    Article  Google Scholar 

  • Rosenbaum, D. A., Meulenbroek, R. J., Vaughan, J., & Jansen, C. (2001). Posture-based motion planning: applications to grasping. Psychological Review, 108(4), 709–734.

    Article  Google Scholar 

  • Rumelhart, D. E., & McClelland, J. L. (1986). Parallel distributed processing: Explorations in the microstructure of cognition: Foundations (parallel distributed processing). Cambridge: MIT Press.

    Google Scholar 

  • Schilling, M. (2009). Dynamic equations in MMC networks: Construction of a dynamic body model. In Proc. of the 12th international conference on climbing and walking robots and the support technologies for mobile machines (CLAWAR).

    Google Scholar 

  • Schilling, M., & Cruse, H. (2007). Hierarchical MMC networks as a manipulable body model. In Proceedings of the international joint conference on neural networks (IJCNN 2007), Orlando, FL (pp. 2141–2146).

    Chapter  Google Scholar 

  • Schilling, M., & Cruse, H. (2008). The evolution of cognition—from first order to second order embodiment. In I. Wachsmuth & G. Knoblich (Eds.), Modeling communication with robots and virtual humans. Berlin: Springer.

    Google Scholar 

  • Schmitz, J., Schneider, A., Schilling, M., & Cruse, H. (2008). No need for a body model: Positive velocity feedback for the control of an 18-dof robot walker. Applied Bionics and Biomechanics, Special Issue on Biologically Inspired Robots, 5(3), 135–147.

    Article  Google Scholar 

  • Shadmehr, R., & Mussa-Ivaldi, F. (1994). Adaptive representation of dynamics during learning of a motor task. Journal of Neuroscience, 14, 3208–3224.

    Google Scholar 

  • Shaw, B. (1903). Man and superman: A comedy and a philosophy. London.

  • Shoemake, K. (1985). Animating rotation with quaternion curves. In SIGGRAPH ’85: Proceedings of the 12th annual conference on computer graphics and interactive techniques (pp. 245–254). New York: ACM Press.

    Chapter  Google Scholar 

  • Smeets, J. B. J., van den Dobbelsteen, J. J., de Grave, D. D. J., van Beers, R. J., & Brenner, E. (2006). Sensory integration does not lead to sensory calibration. Proceedings of the National Academy of Sciences of the United States of America, 103(49), 18,781–18,786. doi:10.1073/pnas.0607687103.

    Article  Google Scholar 

  • Soechting, J., Buneo, C., Herrmann, U., & Flanders, M. (1995). Moving effortlessly in three dimensions: Does Donders’ law apply to arm movement? Journal of Neurosciences, 15(9), 6271–6280.

    Google Scholar 

  • Steels, L. (2003). Intelligence with representation. Philosophical Transactions: Mathematical, Physical and Engineering Sciences, 361(1811), 2381–2395.

    Article  MathSciNet  Google Scholar 

  • Steinkühler, U. (1994). Mmc-modelle zur lösung kinematischer aufgabenstellungen eines redundanten manipulators. Ph.D. thesis, University of Bielefeld.

  • Steinkühler, U., & Cruse, H. (1998). A holistic model for an internal representation to control the movement of a manipulator with redundant degrees of freedom. Biological Cybernetics, 79(6), 457–466.

    Article  Google Scholar 

  • Strauss, R., & Pichler, J. (1998). Persistence of orientation toward a temporarily invisible landmark in drosophila melanogaster. Journal of Comparative Physiology A, 182, 411–423.

    Article  Google Scholar 

  • Stringer, S., & Rolls, E. (2007). Hierarchical dynamical models of motor function. Neurocomputing, 70, 975–990.

    Google Scholar 

  • Uno, Y., Kawato, M., & Suzuki, R. (1989). Formation and control of optimal trajectory in human multijoint arm movement. Biological Cybernetics, 61(2), 89–101.

    Article  Google Scholar 

  • van Beers, R., Wolpert, D., & Haggard, P. (2002). When feeling is more important than seeing in sensorimotor adaptation. Current Biology, 12, 834–837.

    Article  Google Scholar 

  • Wang, L. C. T., & Chen, C. C. (1991). A combined optimization method for solving the inverse kinematics problems of mechanical manipulators. IEEE Transactions on Robotics and Automation, 7(4), 489–499.

    Article  Google Scholar 

  • Wang, X. (1999). Three-dimensional kinematic analysis of influence of hand orientation and joint limits on the control of arm postures and movements. Biological Cybernetics, 80(6), 449–463.

    Article  Google Scholar 

  • Webb, B. (2004). Neural mechanisms for prediction: Do insects have forward models? Trends in Neurosciences, 27(5), 278–282.

    Article  Google Scholar 

  • Whitney, D. E. (1969). Resolved motion rate control of manipulators and human prostheses. IEEE Transactions on Man-Machine Systems, 10, 47–53.

    Article  Google Scholar 

  • Wilson, M. (2002). Six views of embodied cognition. Psychonomic Bulletin & Review, 9(4), 625–636.

    Article  Google Scholar 

  • Wolpert, D., Ghahramani, Z., & Jordan, M. (1995). An internal model for sensorimotor integration. Science, 269, 1880–1882.

    Article  Google Scholar 

  • Wolpert, D., Miall, R., & Kawato, M. (1998). Internal models in the cerebellum. Trends in Cognitive Sciences, 2(9), 338–347.

    Article  Google Scholar 

  • Wolpert, D. M., & Kawato, M. (1998). Multiple paired forward and inverse models for motor control. Neural Networks, 11(7–8), 1317–1329.

    Article  Google Scholar 

  • Yang, A., & Freudenstein, F. (1964). Application of dual-number quaternion algebra to the analysis of spatial mechanisms. ASME Journal of Applied Mechanics, 300–308.

  • Yoshikawa, T. (1985). Manipulability and redundancy control of robotic mechanisms. In Proceedings of the IEEE int. conference on robotics and automation, St. Louis, Missouri (pp. 1004–1009).

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Center of Excellence ‘Cognitive Interaction Technology’ (CITEC), University of Bielefeld, P.O. Box 10 01 31, 33501, Bielefeld, Germany

    Malte Schilling

  2. International Computer Science Institute (ICSI), Berkeley, USA

    Malte Schilling

Authors
  1. Malte Schilling
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Malte Schilling.

Electronic Supplementary Material

Below is the link to the electronic supplementary material.

(PDF 275 KB)

Rights and permissions

Reprints and permissions

About this article

Cite this article

Schilling, M. Universally manipulable body models—dual quaternion representations in layered and dynamic MMCs. Auton Robot 30, 399–425 (2011). https://doi.org/10.1007/s10514-011-9226-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10514-011-9226-3

Keywords

Search

Navigation