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First published online November 17, 2005
Journal of Experimental Biology 208, 4377-4389 (2005)
Published by The Company of Biologists 2005
doi: 10.1242/jeb.01902

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Prediction of kinetics and kinematics of running animals using an analytical approximation to the planar spring-mass system

Justine J. Robilliard¹ and Alan M. Wilson^1,2,*

¹ Structure and Motion Laboratory, The Royal Veterinary College, North Mymms, Hatfield, Hertfordshire AL9 7TA, UK
² Centre for Human Performance, University College London, Brockley Hill, Stanmore, Middlesex, HA7 4LP,

View larger version (13K): [in a new window]   Fig. 1. Diagram of the model. The spring contacts the ground at the origin of the coordinate system (0,0) at some angle 0 to the vertical. At foot contact, represented as the spring on the far left, the point mass (filled circle) has initial horizontal and vertical velocities Vxi and Vyi, respectively. During stance phase the horizontal velocity of the mass decreases to a minimum at midstance and then increases, whilst the spring compresses and then extends as the mass sweeps over the foot, moving from left to right. For an explanation of symbols used, see list.

View larger version (7K): [in a new window]   Fig. 2. Graph (based on the 10° simulation) to show the important features of the horizontal or x position during contact time. The deviation from the position that would result from constant velocity (i.e. a straight line) has been magnified x15 to demonstrate the fluctuation in speed through stance. The gradient of the line at times 0 and Tc are equal and the gradient is at a minimum at midstance.

View larger version (15K): [in a new window]   Fig. 3. (A) Initial horizontal velocity (m s–1) and (B) vertical stiffness (kN m–1) for the analytical solution (squares, red dotted line) and vertical stiffness for the numerical solution (triangles, blue line) against leg angle (degrees).

View larger version (40K): [in a new window]   Fig. 4. Analytical (red dotted lines) and numerical (blue solid lines) solutions for initial leg angles of 15°, 20°, 25°and 30° for (A) x position (m), (B) change in y position (m), (C) change in Vx (m s–1), (D) Vy (m s–1), (E) Ax (m s–2), (F) Ay (m s–2) (G) resultant leg force F (kN) and (H) change in total mechanical energy (ME) against stance time (s) normalised so that midstance occurs at 0 s.

View larger version (39K): [in a new window]   Fig. 5. Percentage differences against leg angle for (A) horizontal position x, (B) vertical position y, (C) horizontal velocity Vx, (D) vertical velocity Vy, (E) horizontal acceleration Ax, (F) vertical acceleration Ay, (G) resultant leg force F and (H) total mechanical energy ME. Note the y-axis scale varies for each subplot. The grey bars represent solutions within the leg angles used by animals.

View larger version (32K): [in a new window]   Fig. 6. (A) Stance time (ms) and (B) initial contact angle (degrees) against horizontal speed (m s–1). Blue diamonds, stance phases from the forelimbs of one Standardbred racehorse (N=1047); red triangles, the computer simulation results.

View larger version (14K): [in a new window]   Fig. 7. Comparison of the analytical approximation (red line) to McMahon and Cheng's outcome for a man running at 18.0 km h–1 (black line; McMahon and Cheng, 1990) for (A) Ay/g against vertical displacement (cm). The dotted line shows the experimental results from Cavagna et al. (1988). (B) Dimensionless horizontal force (Fx/mg) and (C) dimensionless vertical force (Fy/mg) against dimensionless time (0t) where 0=(vertical leg stiffness/m)0.5.

View larger version (11K): [in a new window]   Fig. 8. Comparison of the analytical approximation (red line) to McMahon and Cheng's outcome (solid black line; McMahon and Cheng, 1990) for (A) a trotting dog and (B) a hopping kangaroo. The dotted line shows the experimental results from Cavagna et al. (1988).

View larger version (9K): [in a new window]   Fig. 9. Maximum force (kN) against maximum leg compression (m) for the analytical (squares, red dotted line) and numerical solutions (blue, triangles) for contact angles between 2.5° and 30°. Leg stiffness (the gradient) remains constant for the numerical solutions and is indistinguishable from the stiffness for the analytical solutions.

View larger version (17K): [in a new window]   Fig. 10. Comparison of leg angle between analytical approximations and numerical solutions (A) across the relevant biological range (for initial angles 15°, 20°, 25° and 30°) (blue line, numerical solution is the blue line; red dotted line, analytical solution) and (B) maximum differences in leg angle for all simulations (the grey bars represent the biological range).