Spring-Mass Runner: Leg-Stiffness Gait Dynamics (2D)

Runner Controls

Touchdown angle β 12°

Leg stiffness k·L₀/mg 16

Leg length L₀ 0.90 m

Target speed 2.0 m/s

Active thrust gain 0.85

Playback speed 1.00×

The runner has stumbled — the leg's fixed touchdown angle couldn't get airborne again cleanly, or forward speed decayed to a standstill. Raise the thrust gain (or lower the touchdown angle), then press Reset.

Live Readouts

Running speed

0.00 m/s

Cadence

0 steps/min

Peak vertical GRF

0.00 × BW

Duty factor

How it works

This is the spring-mass runner (SLIP — spring-loaded inverted pendulum), the standard biomechanics template for a running or bouncing gait, and a genuinely different mechanism from the 3D compass-gait walker it pairs with: instead of a rigid leg rotating like a pendulum, the leg here is a real one-sided (push-only) damped spring solved as full 2D Cartesian dynamics, not a single angle:

L  = |hip − foot|,  compression = L₀ − L
Fᵣ = max(0, k·compression − c·dL/dt)      (leg can push, never pull)
a  = (Fᵣ/m)·û − (0, g)                    (û = unit vector foot→hip)

During stance the compressing-then-extending spring stores and returns elastic energy exactly like a tensioned Achilles tendon or a running-blade prosthesis; a small viscous term (c) models the real energy tissue loses to hysteresis on every stride, the running equivalent of the compass gait's heel-strike collision loss. Between stances the leg is unloaded and the body is a true ballistic projectile — the runner is genuinely airborne, unlike a walker.

Two independent active-thrust controls, applied the instant the spring releases at each liftoff, replace what real muscles/ankles (or a powered prosthetic ankle) do — exactly the compass gait's "push-off" idea, just split into its two real components:

vₓ += gain·(targetSpeed − vₓ)              (propulsion: regulates forward speed)
v_y += gain·(√(2g·(apex_target − y)) − v_y) (thrust: regulates hop height)

Touchdown angle β — how far forward the leg is planted; steeper angles brake harder on landing (more stable, shorter stride) but need more thrust to compensate.
Leg stiffness — a stiffer leg stores/returns energy faster (shorter, harder-hitting stance, higher GRF); too soft and the stance drags on and the leg may never fully unload before the next touchdown is due.
Active thrust gain — how completely the active controller replaces each stride's tissue-damping and touchdown-braking losses. Set it too low (as with too little compass-gait push-off) and the runner decelerates and stumbles; the right value settles into an exact repeating stride, visible in the live readouts holding constant.

Real-world relevance: this same compliant-leg template is literally how running-specific prosthetic blades and running-robot legs (e.g. RHex, ATRIAS) are designed and tuned, and how clinicians reason about the elastic-recoil economy of running gait versus the collision-dominated economy of walking gait — the two phenomena the 3D and 2D simulators each model.

Loaded leg (stance) Unloaded leg (flight) Vertical GRF trace