Multibody System Dynamics

, Volume 36, Issue 1, pp 37–65 | Cite as

Optimization-based dynamic prediction of kinematic and kinetic patterns for a human vertical jump from a squatting position

  • Saeed Davoudabadi FarahaniEmail author
  • Michael Skipper Andersen
  • Mark de Zee
  • John Rasmussen


This paper presents the prediction of kinematic and kinetic patterns for human squat jumping using an optimization-based dynamic human movement prediction technique. This method enables prediction of realistic kinematics and kinetics in human squat vertical jumping including muscle and joint forces. The case of vertical jumping is selected because the criterion is clear: to maximize the jump height. First, an anatomically detailed three-dimensional human squat jump model was developed. The movement was then parameterized by means of time functions controlling selected degrees-of-freedom (DOF) of the model. Subsequently, the optimizer found the parameters of these functions to maximize the jump height subject to anatomical and physiological constraints. The results were compared with experimental data from a group of six healthy males. Qualitative and quantitative comparisons between predicted results and experimental observations indicate that the approach is capable of predicting the jump height enhancement in squat vertical jumping with arm swing and reproducing the coordinated motion in terms of kinetics and kinematics.


Musculoskeletal modeling Inverse–inverse dynamics Movement prediction Performance criterion Human squat jump 



This work was supported by the Danish Advanced Technology Foundation.


  1. 1.
    Rasmussen, J., Christensen, S.T., Gföhler, M., Damsgaard, M., Angeli, T.: Design optimization of a pedaling mechanism for paraplegics. Struct. Multidiscip. Optim. 26, 132–138 (2004) CrossRefGoogle Scholar
  2. 2.
    Rasmussen, J., Holmberg, L.J., Sørensen, K., Kwan, M., Andersen, M.S., de Zee, M.: Performance optimization by musculoskeletal simulation. Mov. Sport Sci. 1, 73–83 (2012) CrossRefGoogle Scholar
  3. 3.
    Lemieux, P.O., Tetreault, P., Hagemeister, N., Nuno, N.: Influence of prosthetic humeral head size and medial offset on the mechanics of the shoulder with cuff tear arthropathy: A numerical study. J. Biomech. 46, 806–812 (2013) CrossRefGoogle Scholar
  4. 4.
    Mellon, S.J., Grammatopoulos, G., Andersen, M.S., Pegg, E.C., Pandit, H.G., Murray, D.W., Gill, H.S.: Individual motion patterns during gait and sit-to-stand contribute to edge-loading risk in metal-on-metal hip resurfacing. Proc. Inst. Mech. Eng., H J. Eng. Med. 227, 799–810 (2013) CrossRefGoogle Scholar
  5. 5.
    Weber, T., Dendorfer, S., Dullien, S., Grifka, J., Verkerke, G.J., Renkawitz, T.: Measuring functional outcome after total hip replacement with subject-specific hip joint loading. Proc. Inst. Mech. Eng., H J. Eng. Med. 226, 939–946 (2012) CrossRefGoogle Scholar
  6. 6.
    Grujicic, M., Pandurangan, B., Xie, X., Gramopadhye, A.K., Wagner, D., Ozen, M.: Musculoskeletal computational analysis of the influence of car-seat design/adjustments on long-distance driving fatigue. Int. J. Ind. Ergon. 40, 345–355 (2010) CrossRefGoogle Scholar
  7. 7.
    Rasmussen, J., Tørholm, S., de Zee, M.: Computational analysis of the influence of seat pan inclination and friction on muscle activity and spinal joint forces. Int. J. Ind. Ergon. 39, 52–57 (2009) CrossRefGoogle Scholar
  8. 8.
    Rasmussen, J., Davoudabadi Farahani, S.: Simulating the effect of support points in manual mounting operations. In: International Summit on Human Simulation, Florida, United States (2011) Google Scholar
  9. 9.
    Damiano, D.L., Arnold, A.S., Steele, K.M., Delp, S.L.: Can strength training predictably improve gait kinematics? A pilot study on the effects of hip and knee extensor strengthening on lower-extremity alignment in cerebral palsy. Phys. Ther. 90, 269–279 (2010) CrossRefGoogle Scholar
  10. 10.
    Lampire, N., Roche, N., Carne, P., Cheze, L., Pradon, D.: Effect of botulinum toxin injection on length and lengthening velocity of rectus femoris during gait in hemiparetic patients. Clin. Biomech. 28, 164–170 (2013) CrossRefGoogle Scholar
  11. 11.
    Erdemir, A., McLean, S., Herzog, W., Van den Bogert, A.J.: Model-based estimation of muscle forces exerted during movements. Clin. Biomech. 22, 131–154 (2007) CrossRefGoogle Scholar
  12. 12.
    Pandy, M.G., Zajac, F.E., Sim, E., Levine, W.S.: An optimal control model for maximum-height human jumping. J. Biomech. 23, 1185–1198 (1990) CrossRefGoogle Scholar
  13. 13.
    Zajac, F.E.: Muscle coordination of movement: A perspective. J. Biomech. 26, 109–124 (1993) CrossRefGoogle Scholar
  14. 14.
    Anderson, F.C., Pandy, M.G.: A dynamic optimization solution for vertical jumping in three dimensions. Comput. Methods Biomech. Biomed. Eng. 2, 201–231 (1999) CrossRefGoogle Scholar
  15. 15.
    Anderson, F.C., Pandy, M.G.: Dynamic optimization of human walking. J. Biomech. Eng. 123, 381–390 (2001) CrossRefGoogle Scholar
  16. 16.
    Glitsch, U., Baumann, W.: The three-dimensional determination of internal loads in the lower extremity. J. Biomech. 30, 1123–1131 (1997) CrossRefGoogle Scholar
  17. 17.
    Crowninshield, R.D., Brand, R.A.: A physiologically based criterion of muscle force prediction in locomotion. J. Biomech. 14, 793–801 (1981) CrossRefGoogle Scholar
  18. 18.
    Rasmussen, J., Damsgaard, M., Voigt, M.: Muscle recruitment by the min/max criterion: A comparative numerical study. J. Biomech. 34, 409–415 (2001) CrossRefGoogle Scholar
  19. 19.
    Rasmussen, J., Damsgaard, M., Christensen, S.T.: Inverse–inverse dynamics simulation of musculo-skeletal systems. In: Proceedings of the 12th Conference of the European Society of Biomechanics, Royal Academy of Medicine in Ireland (2000). ISBN 0-9538809-0-7 Google Scholar
  20. 20.
    Abdel-Malek, K., Arora, J.S.: Human Motion Simulation: Predictive Dynamics. Elsevier, Amsterdam (2013) Google Scholar
  21. 21.
    Marler, R.T., Arora, J.S., Yang, J., Kim, H.J., Abdel-Malek, K.: Use of multi-objective optimization for digital human posture prediction. Eng. Optim. 41, 925–943 (2009) CrossRefGoogle Scholar
  22. 22.
    Mi, Z., Yang, J., Abdel-Malek, K.: Optimization-based posture prediction for human upper body. Robotica 27, 607–620 (2009) CrossRefGoogle Scholar
  23. 23.
    Abdel-Malek, K., Mi, Z., Yang, J., Nebel, K.: Optimization-based trajectory planning of the human upper body. Robotica 24, 683–696 (2006) CrossRefGoogle Scholar
  24. 24.
    Ackermann, M., van den Bogert, A.J.: Optimality principles for model-based prediction of human gait. J. Biomech. 43, 1055–1060 (2010) CrossRefGoogle Scholar
  25. 25.
    Kim, J.H., Abdel-Malek, K., Yang, J., Marler, R.T.: Prediction and analysis of human motion dynamics performing various tasks. Int. J. Hum. Factors Model. Simul. 1, 69–94 (2006) CrossRefGoogle Scholar
  26. 26.
    Xiang, Y., Arora, J.S., Rahmatalla, S., Marler, T., Bhatt, R., Abdel-Malek, K.: Human lifting simulation using a multi-objective optimization approach. Multibody Syst. Dyn. 23, 1–21 (2010) zbMATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    Xiang, Y., Arora, J.S., Rahmatalla, S., Abdel-Malek, K.: Optimization-based dynamic human walking prediction: One step formulation. Int. J. Numer. Methods Eng. 79, 667–695 (2009) zbMATHCrossRefGoogle Scholar
  28. 28.
    Xiang, Y., Arora, J.S., Abdel-Malek, K.: Physics-based modeling and simulation of human walking: A review of optimization-based and other approaches. Struct. Multidiscip. Optim. 42, 1–23 (2010) zbMATHCrossRefMathSciNetGoogle Scholar
  29. 29.
    Xiang, Y., Chung, H.J., Kim, J.H., Bhatt, R., Rahmatalla, S., Yang, J., Marler, T., Arora, J.S., Abdel-Malek, K.: Predictive dynamics: An optimization-based novel approach for human motion simulation. Struct. Multidiscip. Optim. 41, 465–479 (2010) zbMATHCrossRefMathSciNetGoogle Scholar
  30. 30.
    Xiang, Y., Arora, J.S., Abdel-Malek, K.: Optimization-based prediction of asymmetric human gait. J. Biomech. 44, 683–693 (2011) CrossRefGoogle Scholar
  31. 31.
    Bobbert, M.F., van Soest, A.J.: Why do people jump the way they do? Exerc. Sport Sci. Rev. 29, 95–102 (2001) CrossRefGoogle Scholar
  32. 32.
    Bobbert, M.F., Casius, L.R., Sijpkens, I.W., Jaspers, R.T.: Humans adjust control to initial squat depth in vertical squat jumping. Int. J. Appl. Phys. 105, 1428–1440 (2008) Google Scholar
  33. 33.
    Blache, Y., Monteil, K.: Effect of arm swing on effective energy during vertical jumping: Experimental and simulation study. Scand. J. Med. Sci. Sports 23, 121–129 (2013) CrossRefGoogle Scholar
  34. 34.
    Damsgaard, M., Rasmussen, J., Christensen, S.T., Surma, E., de Zee, M.: Analysis of musculoskeletal systems in the AnyBody Modeling System. Simul. Model. Pract. Theory 14, 1100–1111 (2006) CrossRefGoogle Scholar
  35. 35.
    Andersen, M.S., Damsgaard, M., Rasmussen, J.: Kinematic analysis of over-determinate biomechanical systems. Comput. Methods Biomech. Biomed. Eng. 12, 371–384 (2009) CrossRefGoogle Scholar
  36. 36.
    Andersen, M.S., Damsgaard, M., MacWilliams, B., Rasmussen, J.: A computationally efficient optimisation-based method for parameter identification of kinematically determinate and over-determinate biomechanical systems. Comput. Methods Biomech. Biomed. Eng. 13, 171–183 (2010) CrossRefGoogle Scholar
  37. 37.
    Rasmussen, J., de Zee, M., Damsgaard, M., Christensen, S.T., Marek, C., Siebertz, K.: A general method for scaling musculo-skeletal models. International Symposium on Computer Simulation in Biomechanics, United States (2005) Google Scholar
  38. 38.
    Delp, S.L.: Surgery simulation: A computer graphics system to analyze and design musculoskeletal reconstructions of the lower limb. Ph.D. thesis, Department of Mechanical Engineering, Stanford University, Stanford, CA (1990) Google Scholar
  39. 39.
    Delp, S.L., Loan, J.P., Hoy, M.G., Zajac, F.E., Topp, E.L., Rosen, J.M.: An interactive graphics-based model of the lower extremity to study orthopaedic surgical procedures. IEEE Trans. Biomed. Eng. 37, 757–767 (1990) CrossRefGoogle Scholar
  40. 40.
    Box, M.: A new method of constrained optimization and a comparison with other methods. Comput. J. 8, 42–52 (1965) zbMATHCrossRefMathSciNetGoogle Scholar
  41. 41.
    Sprague, M.A., Geers, T.L.: Spectral elements and field separation for an acoustic fluid subject to cavitation. J. Comput. Phys. 184, 149–162 (2003) zbMATHCrossRefGoogle Scholar
  42. 42.
    Geers, T.L.: An objective error measure for the comparison of calculated and measured transient response histories. Shock Vib. Bull. 54, 99–107 (1984) Google Scholar
  43. 43.
    Schwer, L.E.: Validation metrics for response histories: Perspectives and case studies. Eng. Comput. 23, 295–309 (2007) CrossRefGoogle Scholar
  44. 44.
    Lund, M.E., de Zee, M., Rasmussen, J.: Comparing calculated and measured curves in validation of musculoskeletal models. In: The XIII International Symposium on Computer Simulation in Biomechanics, Leuven, Belgium (2011) Google Scholar
  45. 45.
    Bobbert, M.F., van Ingen Schenau, G.J.: Coordination in vertical jumping. J. Biomech. 21, 249–262 (1988) CrossRefGoogle Scholar
  46. 46.
    Bobbert, M.F., van Zandwijk, J.P.: Dynamics of force and muscle stimulation in human vertical jumping. Med. Sci. Sports Exerc. 31, 303–310 (1999) CrossRefGoogle Scholar
  47. 47.
    Hara, M., Shibayama, A., Takeshita, D., Fukashiro, S.: The effect of arm swing on lower extremities in vertical jumping. J. Biomech. 39, 2503–2511 (2006) CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2015

Authors and Affiliations

  • Saeed Davoudabadi Farahani
    • 1
    Email author
  • Michael Skipper Andersen
    • 1
  • Mark de Zee
    • 1
    • 2
  • John Rasmussen
    • 1
  1. 1.AnyBody Research Group, Department of Mechanical and Manufacturing EngineeringAalborg UniversityAalborg EastDenmark
  2. 2.Center for Sensory-Motor Interaction, Department of Health Science and TechnologyAalborg UniversityAalborg EastDenmark

Personalised recommendations