Robot Pushing: Motion Cone & Limit Surface

Contact & Push

Contact point on edge 0%

Push angle from normal 0°

Pusher–object friction μ_p 0.35

Push speed 1.0×

Live State

Contact: sticking

Object speed

0.00 u/s

Angular rate ω

0.0°/s

Motion regime

translating

CoR distance

∞

How it works

This models quasi-static non-prehensile pushing (Mason 1986; the ellipsoidal limit-surface approximation used in Lynch & Park's Modern Robotics). A flat pusher contacts one edge of a box resting on a high-friction table. Because the motion is quasi-static, the object's planar twist is (approximately) parallel to the generalized push wrench:

twist  (vx, vy, ω) ∝ (fx, fy, τ / c²)
τ      = px·fy − py·fx           (moment of the push force about the center)
c²     = (w² + h²) / 12          (radius of gyration² of a uniform-pressure rectangle)

The contact itself obeys Coulomb friction: the pusher can only transmit a force inside its own friction cone, half-angle γ = atan(μ_p) around the edge normal. Push outside that cone and the pusher slips along the surface instead of driving the object — so the effective force direction saturates at the cone boundary.

γ = atan(μp)   →   effective angle = clamp(requested angle, −γ, +γ)

Contact point — slides the pushed point along the object's edge, changing the moment arm and hence τ.
Push angle — how far off the edge-normal the pusher force points. Near 0° the box slides almost straight; large angles spin it.
μ_p — widens or narrows the friction cone; a low μ_p clamps most angled pushes, so the box barely rotates.
CoR distance — |v| / |ω|, how far the instantaneous center of rotation sits from the box center (∞ means pure translation).

Real-world relevance: this is exactly the model behind robotic pre-grasp manipulation — bin-picking arms nudge or "singulate" cluttered parts with a single flat pusher before ever closing a gripper, because pushing needs no force-closure grasp at all.

Pushed box Pusher Push force

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