Humanoid Push Recovery Lab: Capture Point Stepping, Ankle Strategy and Cost of Transport Blender 5.2

Linear inverted pendulum (LIPM)

Keep the centre of mass (COM) at constant height z_c and treat the leg as massless. Then x'' = w² (x - p) with w = sqrt(g / z_c) and p the zero-moment point (ZMP). For constant p: x(t) = p + (x0 - p) cosh(wt) + (v0/w) sinh(wt). The same holds independently along the lateral axis, so the 3-D model is two coupled-only-by-footholds LIPMs.

Zero-moment point and the support polygon

The ZMP is the point on the floor where the ground reaction has no horizontal moment. It must stay inside the support polygon (the stance foot, 0.26 x 0.11 m here) or the foot rolls. Shifting it inside the foot (heel to toe) is the ankle strategy; its authority is only about ±0.12 m.

Capture point

xi = x + v / w is where the ZMP would have to be for the robot to come to rest. It obeys xi' = w (xi - p), i.e. it diverges as e^(wt) away from the ZMP while the stable part of the motion decays. Standing push absorbed by the ankle only if xi is inside the foot. Otherwise a step is needed: put the swing foot at the capture point (plus the offset of the nominal gait).

Foot placement rule used here

With step length L and time T the periodic gait has capture-point offset b = L / (e^(wT) - 1) at lift-off. The controller predicts the capture point at touchdown xi_T from the live state and places the foot at p' = xi_T - b (lateral: xi_T - s w_s / (e^(wT) + 1), s = stance side). Replanned every 2 ms during the swing, so a shove mid-step lengthens the step. This removes the unstable mode in one step; the stable mode decays over the following steps.

Limits that make it fall

The foot must land within the leg reach r = sqrt(l² - z_c²) of the COM (leg l = 0.86 m); the COM must also stay within r of the stance foot. The closed form for the largest one-step capturable shove is dv* = w (r_x - L/2) / cosh(w tau) with r_x = sqrt(r² - (w_s/2)²), tau the time left in the step. Shorter step time or lower COM raises the bound; a taller COM or longer step lowers it.

Joint torques, energy, cost of transport

Ground force F = m (w²(x - p), g, w²(y - p_y)) acts at the ZMP. Stance ankle torque is the moment of F about the ankle; knee torque is the moment about the knee hinge of a two-link leg (0.43 + 0.43 m, knee bends forward, IK). The hip torque of the stance leg is zero in the LIPM (F passes through the COM); the hip torque shown is the swing leg (7 kg, 0.55 kg m²). Power = torque x joint rate (filtered finite difference). Cost of transport CoT = E / (m g d). "Ideal net work" is the positive stance-leg power F·v plus swing work; "mechanical" sums |tau w| over joints (no energy recovery); "electrical" adds tau²/Km² with Km = 7 N m / sqrt(W). Humans walk at roughly 0.3 to 0.4; many legged robots are higher.

Independent numerical checks

• Closed form vs an RK4 integrator (error ~1e-12).
• Capture-point law and conserved orbital energy along the trajectory.
• Capture region: a bisection with the ZMP chasing the capture point inside the foot gives the same largest recoverable velocity as x + v/w = toe edge.
• Steady gait: 12 steps repeat exactly (length and speed).
• One-step bound: a shove 10 % below J* is not clamped by reach, 10 % above is.

Simplifications

• Point mass, massless legs, constant COM height, no angular momentum (no arm or torso swing used for balance), no double-support phase, instantaneous heel strike with no impact loss.
• Swing foot follows a smooth path, no foot-ground collision or slipping; no actuator torque or speed limits except the reach.
• Foot-placement is exactly dead-beat; real robots have sensing noise, delays and model error.
• Energy figures are order-of-magnitude, not a prediction for any real machine.