Compass-Gait Walker: Inverted-Pendulum Gait Dynamics

Gait Controls

Step angle α 16°

Push-off gain 0.40

Leg length (body height) 0.90 m

Playback speed 1.00×

The walker has fallen — the stance leg ran out of forward momentum before reaching the next step. Increase push-off or step angle, then press Reset.

Live Readouts

Walking speed

0.00 m/s

Cadence

0 steps/min

Vertical GRF

1.00 × BW

Step length

0.00 m

How it works

This is the compass-gait biped: two rigid, massless legs meeting at a hip point-mass. During single-leg stance the hip swings on a circular arc about the planted foot — a pure inverted pendulum, unstable at the vertical (falling forward is what makes it walk):

θ'' = (g/L) · sin(θ)
θ = stance-leg angle from vertical, L = leg length

Height above the foot is L·cos(θ), maximum at θ=0 — so the hip climbs from heel-strike (θ=−α) to mid-stance (θ=0) and falls from mid-stance to toe-off (θ=+α), reproducing the real vertical bob of a walking body's centre of mass.

At toe-off the swing leg lands and the rigid new leg instantly kills the hip's velocity component along its own length — a real, energy-dissipating collision:

ω⁺ = ω⁻ · cos(2α)      (McGeer's rigid-leg collision result)
R_vertical = M·(g·cos²θ − L·ω²·cosθ)   (axial-leg constraint force)

Left alone, this collision loss brings the walker to a stop within a few steps — exactly the problem a human ankle (or a powered ankle-foot prosthesis) solves with a forward push-off impulse just before heel-strike. The push-off gain slider adds angular velocity at each transition; when it roughly balances the collision loss the walker settles into a steady, repeating gait cycle (a stable limit cycle) — too little and it falls, too much and it keeps accelerating.

Step angle α — half the angle swept by the stance leg each step; sets step length (2L·sinα) and how much energy the collision removes.
Push-off gain — angular velocity injected at heel-strike, standing in for ankle plantarflexion (or a prosthetic push-off actuator).
Vertical GRF readout is the true force-plate quantity clinicians read from a rehab gait lab, in multiples of body weight.

Real-world relevance: this exact passive-dynamics + push-off model is the working tool behind energy-efficient exoskeleton and prosthetic-ankle design and behind understanding why an antalgic or Parkinsonian gait (short α, weak push-off) costs more metabolic energy per metre walked.

Stance leg Swing leg

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