Fig 1.
Human running and the Spring-mass model.
(A) Human stance phases resemble motion of a spring-mass system with no energy loss, alternating with parabolic Flight phases. Body mass is lumped into a single point center of mass (CoM). Traces of (B) vertical ground reaction forces (V. GRF) vs. Time, (C) leg Power vs. Time, (D) and vertical acceleration vs. displacement (V. acc. vs. V. disp.; termed vertical work loop curve) are all shown for both human data (gray lines) and the Spring-mass model (dark solid lines). (E) The energetic Cost of Transport (cost per unit weight and distance) for humans running on slopes (after [11]) is not explained by the Spring-mass model, which only operates at zero ground slope and zero energetic cost. The spring-like behaviors (B-D) should be regarded as pseudo-elastic, because humans and other animals experience dissipation such as in foot-ground collision, and thus, require active muscle actuation.
Fig 2.
Simple running models with and without elastic spring, active actuator, and dissipative elements.
(A) The Spring-mass model comprises a point-mass body and a massless spring for a leg. (B) The Actuator-only model replaces the spring with a massless, active actuator producing extension forces in the leg [14]. (C) The proposed Actuated Spring-mass model combines an actuator and a spring (analogous to a muscle-tendon unit), along with two passive, dissipative elements: a damper in parallel with the spring to model tendon hysteresis, and collision loss to model dissipation of kinetic energy at touchdown. In the models, g is gravitational acceleration and M is body mass. Ll(t), Lt(t), and Lm(t) are time-varying lengths of the leg, spring and actuator, respectively. Parameters k and c are spring stiffness and damping coefficient. Fm(t) is the active actuator force in the leg’s extension direction.
Table 1.
Model parameters and values.
Fig 3.
Running gaits from the Spring-mass (top) and Actuator-only (bottom) models.
They are illustrated by (A) center of mass (CoM) trajectory, (B) vertical ground reaction forces vs. time, (C) leg power performed on the CoM vs. time, and (D) vertical acceleration vs. vertical displacement of the body (or vertical work loop curve). A range of running gaits are shown, varying stiffness k in the Spring-mass model, and the force-rate cost coefficient ε in the Actuator-only model, for a single running speed v (3.5 m s-1) and step frequency f (3 Hz). In the limiting case of infinite spring stiffness or zero force-rate cost, touchdown forces become perfectly impulsive (red arrows).
Fig 4.
Effects of stiff vs. compliant springs on Actuated Spring-mass model minimizing cost of work (with zero force-rate cost).
Optimal running gaits (speed v of 3.0 m s-1) are shown for the Actuated Spring-mass model, including (A) CoM trajectory, (B) vertical ground reaction forces (V. GRF) vs. time, (C) leg power vs. time, and (D) vertical acceleration vs. vertical displacement (V. acc. vs. V. disp.). The model includes passive dissipation (hysteresis and collision), optimized for two spring stiffnesses k (13.7 kN m-1 for compliant and 109.5 kN m-1 for stiff). The stiffer spring yields a more vertical leg, shorter stance time and bouncier gait, with higher peak forces and leg power. Net actuator work is similar in both cases.
Fig 5.
Actuator work and power as a function of spring stiffness and running speed in the Actuated Spring-mass model, minimizing cost of work (with zero force-rate cost).
(A) Work vs. stiffness for speeds of 2.5–3.5 m s-1. Shown are active Actuator work (black), Collision work magnitude (red), and Hysteresis work magnitude (blue). Spring diagrams (inset) illustrate touchdown angles for each stiffness. (B) Power vs. time for very compliant and very stiff springs, for each running speed. Shown are net Leg power (black lines), Spring power (orange shaded area), Actuator power (blue shaded area). Results are for spring stiffness ranging 13.7–109.5 kN m-1.
Fig 6.
Effect of work and force-rate costs on running using Actuated Spring-mass model.
(top:) Vertical CoM displacement vs. time, vertical GRF vs. time, and leg Power vs. time, for varying force-rate cost coefficient ϵ. (bottom:) Actuator work cost EW (thin blue line), Total cost EW + ER (work and force rate, solid black line), and Force-rate cost ER (difference between lines) vary with the coefficient. Impulsive actions (red arrows, V. GRF and leg Power) occur at Collision, and overall leg power (solid black line) includes contributions from the Spring and Actuator. All solutions are shown for v of 3 m s-1 and f of 2.94 Hz.
Fig 7.
Comparison of Unified Actuated Spring-mass model (top) including work and force-rate costs against human data (bottom). Shown are (A) CoM trajectories, (B) vertical ground reaction forces, (C) leg mechanical power, and (D) vertical acceleration vs. displacement. Initial force transients are highlighted (red impulse arrow for model, red line for human). In (C), spring (orange shaded area) and actuator work (blue shaded area) contributions are shown. Gait parameters v and f are 3.9 m s-1 and 3 Hz, respectively. Stiffness and force-rate coefficient in the model are selected to approximately match stance time duration: k* is 35.6 kN m-1 and ε* is 0.5∙10−3.
Fig 8.
Unified model energy cost vs. speed and spring stiffness, including force-rate cost.
(A) Energetic cost per time versus speed is shown for the Actuated Spring-mass model (k* of 35.6 kN m-1, ε* of 0.5∙10−3; red curve) and for empirical metabolic data of human subjects running on a treadmill (mean ± standard deviation; [48]). Model cost includes costs for work and force-rate, plus a constant offset associated with human resting metabolism (dashed horizontal line). (B) The model’s energetic cost is shown for three speeds v (2.5–3.5 m s-1) and with parameter variation of spring stiffnesses k (13.7–109.5 kN m-1), with total cost (black lines), force-rate cost (difference between offset and magenta lines), and actuator work cost (difference between total and magenta lines). The unified model’s spring nominal stiffness k* is indicated (red line in A, red symbol in B).
Fig 9.
Energetic cost of running vs. ground slope for unified Actuated Spring-mass model.
(A) Model Cost of Transport (solid line) compared to humans (circles; [11]). Also shown are asymptotes (thin lines) for muscle efficiency of positive and negative mechanical work (25% and -120%, respectively). (B) Contributors to model Cost of Transport: positive work cost, negative work cost, force-rate cost ER, and a constant offset. Parameter sensitivities are included for varying (C) stiffness k, (D) force-rate coefficient ε, (E) hysteresis (damping ratio ζ), and (F) collision fraction CF. Each trace indicates variation from lowest to highest parameter values: k ranging 13.7 kN m-1 –∞, ε ranging 0–2∙10−3, ζ ranging 0–0.2, CF ranging 0–0.06. All model results are for nominal running at speed of 3 m s-1 and step frequency of 2.94 Hz.