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. L l (t) , L t (t) , and L m (t) are time-varying lengths of the leg, spring and actuator, respectively. Parameters k and c are spring stiffness and damping coefficient. F m (t) is the active actuator force in the leg’s extension direction.