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Application of motor algebra to the analysis of human arm movements

Published online by Cambridge University Press:  01 July 2008

Sigal Berman ,
Dario G. Liebermann  and
Tamar Flash
Sigal Berman*
Affiliation:
Department of Industrial Engineering and Management, Ben-Gurion University of the Negev, Beer-Sheva, Israel Department of Computer Science and Applied Mathematics, Weizmann Institute of Science, Rehovot, Israel
Dario G. Liebermann
Affiliation:
Department of Physical Therapy, Tel-Aviv University, Ramat-Aviv, Israel
Tamar Flash
Affiliation:
Department of Computer Science and Applied Mathematics, Weizmann Institute of Science, Rehovot, Israel
*
*Corresponding author. E-mail: [email protected]
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Summary

Motor algebra, a 4D degenerate geometric algebra, offers a rigorous yet simple representation of the 3D velocity of a rigid body. Using this representation, we study 3D extended arm pointing and reaching movements. We analyze the choice of arm orientation about the vector connecting the shoulder and the wrist, in cases for which this orientation is not prescribed by the task. Our findings show that the changes in this orientation throughout the movement were very small, possibly indicating an underlying motion planning strategy. We additionally examine the decomposition of movements into submovements and reconstruct the motion by assuming superposition of the velocity profiles of the underlying submovements by analyzing both the translational and rotational components of the 3D spatial velocity. This movement decomposition method reveals a larger number of submovement than is found using previously applied submovement extraction methods that are based only on the analysis of the hand tangential velocity. The reconstructed velocity profiles and final orientations are relatively close to the actual values, indicating that single-axis submovements may be the basic building blocks underlying 3D movement construction.

Keywords

Geometric algebraMotor algebraArm motionSpatial velocity
Type
Article
Information
Robotica , Volume 26 , Special Issue 4: Geometry in Robotics: Sensing , July 2008 , pp. 435 - 451
Copyright
Copyright © Cambridge University Press 2007

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References

1.Tweed, D. and Vilis, T., “Implications of rotational kinematics for the oculomotor system in three dimensions,” J. Neurophysiol. 58 (4), 832–849 (1987).Google Scholar
2.Hestenes, D., “Invariant body kinematics: I. Saccadic and compensatory eye movements,” Neural Netw. 7 (1), 65–77 (1994).CrossRefGoogle Scholar
3.Murray, R. M., Li, Z. and Sastry, S. S., A Mathematical Introduction to Robotic Manipulation (CRC Press, Boca Raton, FL, 1994).Google Scholar
4.Hestenes, D. and Fasse, E., “Homogeneous Rigid Body Mechanics with Elastic Coupling,” In: Applications of Geometric Algebra with Applications in Computer Science and Engineering (Doerst, L., Doran, C. and Lasenby, J., Eds.) (Birkhauser, Boston, MA, 2002) pp. 3–34.Google Scholar
5.Mason, M. T., Mechanics of Robotic Manipulation (MIT Press, Cambridge, MA, 2001).Google Scholar
6.Bernstein, N., The Coordination and Regulation of Movements (Pergamon Press, Oxford, UK, 1967).Google Scholar
7.Flash, T. and Hogan, N., “The coordination of arm movements: An experimentally confirmed mathematical model,” J. Neurosci. 5 (6), 1688–1703 (1985).Google Scholar
8.Uno, Y., Kawato, M. and Suzuki, R., “Formation and control of optimal trajectory in human multijoint arm movement. Minimum torque-change model,” Biol. Cybern. 61, 89–101 (1989).Google Scholar
9.Rohrer, B. and Hogan, N., “Avoiding spurious submovement decomposition: A globally optimal algorithm,” Biol. Cybern. 89, 190–199 (2003).Google Scholar
10.Pasalar, S., Roitman, A. V. and Ebner, T. J., “Effects of speed and force fields on submovements during circular manual tracking in humans,” Exp. Brain Res. 163, 214–225 (2005).Google Scholar
11.Dounskaia, N., Wisleder, D. and Johnson, T., “Influence of biomechanical factors on substructure of pointing movements,” Exp. Brain Res. 164, 505–516 (2005).CrossRefGoogle ScholarPubMed
12.Meyer, D. E., Kornblum, S., Abrams, R. A., Wright, C. E. and Smith, J. E. K., “Optimality in human motor preformance: Ideal control of rapid aimed movements,” Psychol. Rev. 95 (3), 340–370 (1988).Google Scholar
13.Novak, K. E., Miller, L. E. and Houk, J. C., “The use of overlapping submovements in the control of rapid hand movements,” Exp. Brain Res. 144, 351–364 (2002).Google Scholar
14.Flash, T. and Henis, E., “Arm trajectory modification during reaching towards visual targets,” J. Cogn. Neurosci. 3, 220–230 (1991).Google Scholar
15.Fishbach, A., Roy, S. A., Bastianen, C., Miller, L. E. and Houk, J. C., “Deciding when and how to correct a movement: discrete submovements as a decision making process,” Exp. Brain Res. 177, 45–63 (2007).Google Scholar
16.Rohrer, B. and Hogan, N., “Avoiding spurious submovement decomposition II: A scattershot algorithm,” Biol. Cybern. 94, 409–414 (2006).Google Scholar
17.Georgopoulos, A. P., Kalaska, J. F. and Massey, J. T., “Spatial trajectories and reaction times of aimed movements: Effects of practice, uncertainty and change in target location,” J. Neurophysiol. 46, 725–743 (1981).CrossRefGoogle ScholarPubMed
18.Soechting, J. F. and Lacquaniti, F., “Modification of trajectory of a pointing movement in response to a change in target location,” J. Neurophysiol. 49, 548–564 (1983).CrossRefGoogle ScholarPubMed
19.Henis, E. and Flash, T., “Mechanisms underlying the generation of averaged modified trajectories,” Biol. Cybern. 72, 407–419 (1995).Google Scholar
20.Plotnik, M., Flash, T., Inzelberg, R., Schechtman, E. and Korczyn, A., “Motor switching abilities in Parkinson's disease and old age: Temporal aspects,” J. Neurol. Neurosurg. Psychiatry 65, 328–337 (1998).CrossRefGoogle ScholarPubMed
21.Gat-Falik, T. and Flash, T., “The super position strategy for arm trajectory modification in robotic manipulation,” IEEE Trans. Syst. Man Cybern. Part B 29 (1), 83–95 (1999).CrossRefGoogle Scholar
22.Bhushan, N. and Shadmehr, R., “Computational nature of human adaptive control during learning of reaching movements in force fields,” Biol. Cybern. 81, 39–60 (1999).Google Scholar
23.Soechting, J. F., Buneo, C. A., Herrmann, U. and Flanders, M., “Moving effortlessly on three dimensions: Does Donders' law apply to arm movements?,” J. Neurosci. 15, 6271–6280 (1995).CrossRefGoogle Scholar
24.Biess, A., Flash, T. and Liebermann, D. G., “Multijoint point-to point arm movements of humans in 3d-space—minimum kinetic energy paths,” [R. Meulenbroek and B. Steenberger, Eds.] Proceedings of the 10th Biennial Conference of the International Graphonomics Society (2001) pp. 142–147.Google Scholar
25.Wang, X. and Verriest, J. P., “A geometric algorithm to predict the arm reach posture for computer-aided ergonomic evaluation,” J. Vis. Comput. Anim. 9, 33–47 (1998).Google Scholar
26.Straumann, D., Haslwanter, T., Hepp-Reymond, M. C. and Hepp, K., “Listing's law for the eye, head and arm movements and their synergistic control,” Exp. Brain Res. 86, 209–215 (1991).Google Scholar
27.Miller, L. E., Theeuwen, M. and Gielen, C. C. A. M., “The control of arm pointing movements in three dimensions,” Exp. Brain Res. 90, 415–426 (1992).Google Scholar
28.Liebermann, D. G., Biess, A., Friedman, J., Gielen, C. C. A. M. and Flash, T., “Intrinsic joint kinematic planning-I; Reassessing the Listing's law constraint in the control of three-dimensional arm movements” Exp. Brain Res. 171, 139–154 (2006).Google Scholar
29.Handzel, A. A. and Flash, T., “The Geometry of Eye Rotations and Listing's Law,” In Advances in Neural Information Processing Systems (Touretzky, D., Mozer, M. and Hasselmo, M., Eds.), (MIT Press, Cambrige, MA, 1996), vol. 8 pp. 117–123.Google Scholar
30.Friedman, J., The Planning of Three Dimensional Fully Extended Arm Pointing Movements, M.Sc. Dissertation (Rehovot, Israel: The Weizmann Institute of Science 2002).Google Scholar
31.Minken, A. W. H., Van Opstal, A. J. and Van Gisbergen, J. A. M., “Three-dimensional analysis of strongly curved saccades elicited by double step stimuli,” Exp. Brain Res. 93, 521–533 (1993).Google Scholar
32.Lasenby, J., Lasenby, A. N. and Doran, C. J. L., “A unified mathematical language for physics and engineering in the 21st century,” Phil. Trans. R. Soc. Lond. 21–39 (1996).Google Scholar
33.Hestenes, D., New Foundations for Classical Mechanics, 2nd ed. (Kluwer Academic, Dordrecht, The Netherlands, 1999).Google Scholar
34.Bayro-Corrochano, E., Daniilidis, K. and Sommer, G., “Motor algebra for 3D kinematics: The case of the hand–eye calibration,” J. Math. Imaging Vis. 13, 79–100 (2000).Google Scholar
35.Selig, J. M., Geometric Fundamentals of Robotics, 2nd ed. (Springer Science, New York, 2005).Google Scholar
36.Gull, S., Lasenby, A. and Doran, C., “Imaginary numbers are not real—the geometric algebra of spacetime,” Found. Phys. 23 (9), 1175–1196 (1993).Google Scholar
37.Bayro-Corrochano, E. and Kahler, D., “Motor algebra approach for computing the kinematics of robot manipulators,” J. Robot. Syst. 17 (9), 495–516 (2000).Google Scholar
38.Bayro-Corrochano, E., “Motor algebra approach for visually guided robotics,” Pattern Recognit. 35, 279–294 (2002).Google Scholar
39.Bayro-Corrochano, E., “Modeling the 3D kinematics of the eye in the geometric algebra framework,” Pattern Recognit. 36 (12), 2993–3012 (2003).Google Scholar
40.Hiniduma, S. S., Gamage, U. and Lasenby, J., “New least squares solutions for estimating the average centre of rotation and axis of rotation,” J. Biomech. 35, 87–93 (2002).Google Scholar
41.Lee, D., Port, N. L. and Goergopoulos, A. P., “Manual interception of moving targets II. One line control of overlapping submovements,” Exp. Brain Res. 116, 421–433 (1997).CrossRefGoogle Scholar
42.Hestenes, D., “Invariant body kinematics: II. Reaching and Neurogeometry,” Neural Netw. 7 (1), 79–88 (1994).CrossRefGoogle Scholar
43.Plamondon, R., “A Theory of Rapid Movements,” In: Tutorials in Motor Behavior (G. E. Stelmach and J. Requin, Eds) (Elsevier, Amsterdam, The Netherlands, 1992), vol. II, pp. 55–69.Google Scholar
44.Plamondon, R., Alimi, A. M., Yergeau, P. and Leclerc, F., “Modelling velocity profiles of rapid movement: A comparative study,” Biol. Cybern. 69, 119–128 (1993).Google Scholar
45.Schmidt, R. A. and Lee, T. D., Motor Control and Learning: A Behavioral Emphasis, 4th ed. (Human Kinetics, Champaign, IL, 2005).Google Scholar