Unilateral contact, friction, and compliant models

Learning outcomes

• Formulate ideal unilateral contact and a friction cone.
• Distinguish forces from impact impulses.
• Compare rigid, regularized, and hydroelastic contact assumptions.

Prerequisites: Linear algebra, ODEs, and earlier lectures in this sequence.

Unilateral contact

An ideal nonadhesive normal contact can push but cannot pull. At position level, a basic complementarity statement is

An impact changes velocity over a short interval. Its impulse p has units N s, whereas force has units N. A discrete solver's impulse can be related to an average force over a timestep, but they are not the same quantity.

Coulomb friction

The ideal friction cone satisfies $\|f_t\|\le\mu f_n$. During sliding, friction opposes tangential velocity and lies on the boundary. During sticking, an interior force can maintain zero slip. The direction at exactly zero slip needs a constraint formulation; normalizing a zero tangential velocity is not a valid implementation.

A pyramidal approximation replaces the circular cone with linear inequalities. It changes the feasible force set and can introduce directional artifacts.

Worked sliding block

For a mass sliding horizontally with normal force approximately mg, Coulomb deceleration has magnitude $\mu g$. With initial speed 1 m/s and $\mu=0.2$, the ideal stopping distance is $1/(2\mu g)\approx0.255$ m. This assumes continuous contact, constant friction, no rolling, and no other forces.

A sphere can roll, so using it for this sliding-only benchmark invalidates the assumptions. Practice 6 uses a constrained translating body to isolate sliding friction and labels that simplification.

Compliant and regularized contact

MuJoCo uses a regularized contact formulation rather than enforcing the simple ideal complementarity model exactly. Penetration and slip can reflect chosen compliance, timestep, and solver behavior. Vary one factor at a time.

Drake's hydroelastic model represents contact using distributed pressure over a contact surface Drake hydroelastic. It is a different modeling approach, especially relevant to distributed contact patches. This course compares assumptions; it does not claim matching numerical trajectories across the two models.

Practice

Measure maximum penetration, resting force, and tangential stopping distance. Change friction separately from solref and timestep. A smaller penetration is not automatically a more physically accurate result if no measured compliance is available.

Exercise: A body at rest on a plane has a nonzero normal force but zero contact power. Is that consistent? Answer: yes. Mechanical power is force dotted with contact-point velocity. A support force can be nonzero while doing no work in ideal sticking contact.

The contact experiment is intentionally small enough that every reported force can be checked against a free-body diagram.