Continuous Collision (CCD)

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Continuous Collision Detection (CCD & Swept Volumes): In discrete physics simulations, collision detection evaluates positions only at discrete time steps $t$ and $t + \Delta t$. A high-speed projectile (or robotic arm) can move entirely through a thin wall in a single frame without either endpoint overlapping the obstacle—a catastrophic glitch known as tunneling. CCD treats motion as a 4D spatiotemporal swept volume (e.g. a capsule) to compute the exact continuous Time of Impact (TOI).

CCD Swept Volume Principles:

1. Swept Capsule Primitive: A moving sphere sweeps a 3D capsule volume between $t_0$ and $t_1$. 2. Time of Impact (TOI): Solves root $f(t) = 0$ for the exact fractional timestamp $t \in [0, 1]$ of first contact. 3. Zero Tunneling Guarantee: Fast-moving mechanical parts cannot pass through safety barriers.

python

1def solve_ccd_time_of_impact(p0, p1, radius, wall_x):
2    """
3    Continuous Collision Detection (CCD) for high-speed sphere vs vertical wall.
4    Prevents discrete tunneling where object jumps across wall between frames.
5    Solves for exact Time of Impact (TOI) in interval [0, 1].
6    """
7    dx = p1[0] - p0[0]
8    if abs(dx) < 1e-6:
9        return None  # Parallel motion
10        
11    # Hit occurs when center reaches wall_x - radius
12    toi = (wall_x - radius - p0[0]) / dx
13    if 0.0 <= toi <= 1.0:
14        hit_pos = p0 + toi * (p1 - p0)
15        return toi, hit_pos
16    return None

Projectile Velocity (Step Length)

4.50

2 min read•1 page

Speculative Contacts (Constraint-Based Anti-Tunneling): Iterative continuous collision solvers (like conservative advancement) require expensive root-finding loops that drag down simulation framerates. Speculative contacts solve tunneling directly inside the linear velocity constraint solver. By generating a "speculative" contact joint before the bodies actually touch, the velocity solver clamps closing speeds so the object arrives flush against the obstacle without ever piercing it.

Speculative Contact Mechanism:

1. Predictive Distance Filter: Detects impending collision within time step d < v_n Δt. 2. One-Way Impulse Limit: Clamps approaching relative velocity to v_rel ≤ d / Δt. 3. Zero Solver Iteration Overhead: Integrates seamlessly into sequential impulse LCP solvers (e.g. Box2D, Bullet).

python

1def generate_speculative_contact(pos, vel, dt, wall_pos, wall_normal):
2    """
3    Speculative Contact generation (Erwin Coumans / Bullet Physics).
4    If distance to obstacle d is less than incoming velocity component along normal:
5    d < (vel . wall_normal) * dt,
6    insert a speculative contact constraint with distance offset d.
7    Limits normal impulse so body halts exactly at contact plane without penetrating.
8    """
9    dist = np.dot(wall_pos - pos, wall_normal)
10    v_proj = -np.dot(vel, wall_normal)
11    
12    if 0.0 < dist < v_proj * dt:
13        # Generate speculative contact constraint before penetration occurs
14        max_impulse_v = dist / dt
15        return True, max_impulse_v
16    return False, 0.0

Approaching Speed v

3.00