Coefficient of Restitution (COR)

The Coefficient of Restitution (COR) is formally defined as the negative ratio of the relative separation velocity to the relative approach velocity of two bodies along the line of impact. For a collision between bodies A and B:

COR = - (v_A2 - v_B2) / (v_A1 - v_B1)

Where v_A1, v_B1 are pre-impact velocities and v_A2, v_B2 are post-impact velocities. This dimensionless value quantifies the elasticity of the collision.

The COR's scalar range has direct physical meaning:

• COR = 1: A perfectly elastic collision. Kinetic energy is conserved; bodies rebound with reversed relative velocity. Idealized case (e.g., superball).
• COR = 0: A perfectly inelastic collision. Maximum kinetic energy loss; bodies stick together post-impact (e.g., clay).
• 0 < COR < 1: A partially elastic collision. Most real-world impacts (e.g., baseball on bat, robot gripper on object). Energy is dissipated as heat, sound, or deformation.

COR is not a pure material property. It depends on:

• Material Pair: The combination of the two colliding materials (steel-on-steel vs. rubber-on-concrete).
• Impact Velocity: COR often decreases with increasing impact speed due to greater plastic deformation.
• Geometry & Temperature: Shape, surface roughness, and temperature affect the effective COR. In simulation, a constant COR is often used, but high-fidelity models may implement velocity-dependent COR tables for greater accuracy.

Accurate COR modeling is vital for sim-to-real transfer in robotics. An incorrect COR in simulation leads to:

• Divergent Dynamics: A simulated robot will learn policies based on incorrect rebound behavior.
• Reality Gap: Policies fail on physical hardware where real-world damping and material properties differ. Domain randomization of COR during training is a common technique to build robust policies that can handle a range of real-world physical interactions.

The COR directly determines the kinetic energy lost in a collision. For a central impact, the fractional kinetic energy loss is:

Energy Loss = 1 - COR²

This quadratic relationship shows that a small decrease in COR causes a significant increase in energy dissipation. A COR of 0.8 results in a 36% energy loss. This is crucial for simulating systems where damping and friction (Coulomb friction) are coupled with collision response.

In physics engines like MuJoCo, Bullet, or Isaac Sim, the COR is a key parameter in the contact model. It is typically applied in the constraint solver or impulse-based dynamics resolution step:

• The solver calculates a normal impulse to prevent penetration.
• The COR scales this impulse to achieve the correct post-impact relative velocity.
• It works in conjunction with the friction cone model for tangential forces. Incorrect solver ordering between friction and restitution can cause visual artifacts like jitter.

The computational process of identifying when two or more simulated objects intersect. COR is applied after collision detection provides the contact points and normals. The pipeline is:

• Broad Phase: Quickly filters non-colliding object pairs using bounding volumes.
• Narrow Phase: Computes precise contact details (point, normal, penetration depth).
• Resolution: Applies impulses based on physics properties, including COR, to separate bodies.

The force generated at the interface of two colliding bodies. In impulse-based resolution, the COR directly scales the normal component of the contact impulse. Key components:

• Normal Force: Acts perpendicular to the contact surface, repelling bodies. Magnitude is influenced by COR.
• Tangential (Friction) Force: Acts parallel to the surface, governed by a separate friction model (e.g., Coulomb). A COR of 1 results in a perfectly elastic normal impulse, while 0 results in an inelastic, dampening impulse.

A simulation method that resolves collisions via instantaneous changes in velocity (impulses) rather than continuous forces. COR is central to its collision response formula: J = -(1 + e) * (v_rel · n) / (1/m_a + 1/m_b + ...) Where J is the impulse magnitude, e is the COR, v_rel is relative velocity, and n is the contact normal. This method is computationally efficient for real-time applications like games and interactive robotics simulators.

A mathematical framework used in advanced physics engines to solve for contact and friction forces simultaneously. COR is embedded within the LCP formulation as a restitution parameter that defines the relationship between pre- and post-impact velocities. The solver finds impulses that satisfy:

• Non-penetration constraints.
• Coulomb friction cone constraints.
• The velocity-based restitution law (v_sep >= -e * v_app). This provides a more unified and stable solution for complex, multi-contact scenarios.

An alternative constraint enforcement technique where intersecting objects are pushed apart by a spring-damper force proportional to penetration depth. COR is implicitly modeled by the damping coefficient of the contact spring.

• High Damping: Simulates a low COR (inelastic collision).
• Low Damping: Simulates a high COR (elastic collision). While simpler to implement, the penalty method can suffer from stability issues with stiff springs and does not directly enforce the velocity-based definition of COR.

The classical model for dry friction tangential to a contact surface. It operates independently of the COR, which governs the normal direction. The model defines:

• Static Friction: Maximum force before sliding begins: |f_t| ≤ μ_s * |f_n|.
• Kinetic Friction: Constant force during sliding: |f_t| = μ_k * |f_n|. In a collision, the normal impulse (scaled by COR) determines f_n, which then caps the available friction force f_t. This separation is critical for simulating realistic object behavior like a bouncing, spinning ball.