Robot Crowd Navigation 2D — Reciprocal Collision Avoidance (ORCA)

Crowd & Robot

Crowd density 35

Robot patrol speed 1.40

Sensing radius 4.5 m

Robot yield share 50%

Avoidance time horizon τ 2.0 s

Display

Sensing radius ring Velocity vectors Robot's active constraint lines

Live Readouts

Robot speed

0.00 m/s

Nearest pedestrian

— m

Active constraints

Near-misses

How it works

Every agent (the robot and each pedestrian) picks a preferred velocity toward its own goal. To avoid collisions, each pair of nearby agents A, B computes a truncated velocity obstacle: the set of relative velocities that would bring their circular bodies (radius r_A+r_B) within a time horizon τ.

relPos = posB − posA
relVel = velA − velB
w = relVel − relPos/τ
if |w| escapes the truncation cone:
  u = (minimal vector to leave the cone) − relVel
else:
  u = projection onto the nearest cone leg

This is the ORCA (Optimal Reciprocal Collision Avoidance) construction, following van den Berg et al. 2011. Each agent then takes only a share of the fix, u·share, added to its own velocity — the other agent implicitly takes the rest, so both sides change course a little instead of one dodging entirely. The half-plane {v : (v − (vel+u·share))·n ≥ 0} becomes a linear constraint; a few passes of projecting the preferred velocity onto every active constraint (Gauss–Seidel relaxation) converge to a collision-free velocity, which is then clamped to the agent's max speed.

Sensing radius — only neighbors inside this distance generate a constraint (a real robot's LIDAR/camera range).
Robot yield share — the fraction of the avoidance vector u the robot itself absorbs against a pedestrian; 0 = pedestrians always give way, 1 = the robot always yields, 0.5 = textbook reciprocal ORCA.
Time horizon τ — how far ahead the robot "looks" for a future collision; a small τ reacts late and abruptly, a large τ reacts to distant, still-uncertain futures and produces gentler, earlier course changes.
Pedestrian–pedestrian pairs always split 50/50, which is what keeps the crowd itself collision-free.

This 2D top-down view renders the same planar agent positions the 3D version moves in space — the underlying ORCA solve is identical, only the camera changes. Real-world relevance: this exact velocity-obstacle math (RVO/ORCA) is what lets event-security and logistics robots move through dense, unpredictable crowds at concerts, stadiums and exhibitions without freezing up or shoving people — the same technique underlies multi-drone swarms and warehouse AMR fleets.

Patrol robot Pedestrians Sensing radius

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