Abstract
A three-rigid-links model is constructed for a gymnast performing a kip-up maneuver on a horizontal bar. Equations of motion with constrained, voluntary torques at hip and shoulder joints give a well-posed optimal control problem when boundary conditions and a performance criterion for the maneuver are specified. An approximate numerical solution for the minimum-time performance of this nonlinear process is obtained by the method of steepest descent. Results of the computations are compared with experimental results. Difficulties of solving human motion problems by existing numerical methods are pointed out.
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Abbreviations
- element 1:
-
arm system
- element 2:
-
head-neck-torso system
- element 3:
-
leg system
- ϕ :
-
angle between element 1 and vertical
- θ :
-
angle between elements 1 and 2
- ψ :
-
angle between elements 2 and 3
- O 1 :
-
hinge axis between elements 1 and 2
- O 2 :
-
hinge axis between elements 2 and 3
- O 3 :
-
hinge axis representing fist-horizontal-bar system
- T 1 :
-
torque between elements 1 and 2
- T 2 :
-
torque between elements 2 and 3
- l 1 :
-
distance betweenO 3 andO 1
- l 2 :
-
distance betweenO 1 andO 2
- I i :
-
moment of inertia of elementi about its CG about an axis perpendicular to the plane of motion,i = 1,2,3
- I r :
-
moment of inertia of the horizontal bar about its longitudinal axis
- m i :
-
mass of elementi, i=1,2,3
- r 1 :
-
distance between O3 and CG of element 1
- r 2 :
-
distance between O1 and CG of element 2
- r 3 :
-
distance between O1 and CG of element 3
- g :
-
acceleration due to gravity
References
Smith, P. G., andKane, T. R.,On the Dynamics of the Human Body in Free Fall, Journal of Applied Mathematics, Vol. 35, pp. 167–168, 1968.
Hanavan, E. P.,A Personalized Mathematical Model of the Human Body, Aerospace Medical Research Laboratory, AMRL-TR-102, 1964.
Samras, R. K.,Muscle Torque Measurements, Efforts on Optimal Motion and Human Performance Evaluation, University of Florida, MS Thesis, 1971.
Ghosh, T. K.,Dynamics and Optimization of a Human Motion Problem, University of Florida, PhD Thesis, 1974.
Bryson, A. E., andDenham, W. F.,Optimal Programming Problems with Inequality Constraints II: Solution by Steepest Ascent, AIAA Journal, Vol. 12, pp. 25–34, 1964.
Hermes, H.,Controllability and Singular Problems, SIAM Journal on Control, Vol. 2, pp. 241–260, 1965.
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Communicated by J. V. Breakwell
This research was partially supported by the National Science Foundation through Grants Nos. GK-4944 and GK-37024x.
Appreciation of Dr. Tom Bullock for discussion on numerical optimization techniques and Mr. Tom Boone for his services as an experimental test subject is gratefully acknowledged.
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Ghosh, T.K., Boykin, W.H. Analytic determination of an optimal human motion. J Optim Theory Appl 19, 327–346 (1976). https://doi.org/10.1007/BF00934100
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DOI: https://doi.org/10.1007/BF00934100