Robot Contact and Friction: Stick or Slide

Why Does This Exist?

A foot supports a robot vertically but slips when the controller requests too much horizontal force. Touching the ground does not supply arbitrary force. Contact and friction models specify which forces can exist and how they change when bodies separate, stick or slide. Those distinctions matter before optimizing any motion.

The contact normal points outward from the supporting surface into the supported body. Tangential directions lie along that surface. Forces change acceleration through robot dynamics; the contact law restricts which external forces are available. This page teaches force admissibility and slip transitions. Robot simulation explains how numerical stepping and contact solvers approximate their evolution.

Think of It Like This

How It Actually Works

Which forces are allowed at rest?

For a nonadhesive contact, normal force $f_n$ is nonnegative: the surface pushes but does not pull. If signed separation $\phi$ is positive, normal force is zero. At closed contact, $\phi=0$ and normal force may be positive. In an ideal rigid model, $\phi\geq0$, $f_n\geq0$, and $\phi f_n=0$. These are compatibility conditions, not a formula computing force from gap. The body dynamics and other contacts also participate in finding the force.

Under Coulomb static friction,

$\|\mathbf f_t\|_2\leq\mu_s f_n,$

where $\mathbf f_t$ is the tangential force vector in newtons and $\mu_s$ is the dimensionless static coefficient. In a plane this reduces to $|f_t|\leq\mu_s f_n$. The cone half-angle is $\arctan\mu_s$. Inside that set, static friction takes the value required for sticking; it does not automatically equal the boundary value. Modern Robotics' friction chapter distinguishes the set of possible zero-slip forces from the sliding rule.

What changes after slip starts?

For nonzero tangential relative velocity $\mathbf v_t$, one kinetic approximation is

$\mathbf f_t=-\mu_k f_n\frac{\mathbf v_t}{\|\mathbf v_t\|_2},$

with kinetic coefficient $\mu_k$. The minus sign opposes relative sliding, not necessarily the body's world velocity. Its contact power is $\mathbf f_t\cdot\mathbf v_t=-\mu_k f_n\|\mathbf v_t\|_2\leq0$: friction dissipates energy at the interface. Do not divide by zero at rest; select the sticking condition or use an explicitly justified regularization.

Compute the force-mode change

Take a stationary 2 kg block on a horizontal floor. Let gravity be 9.81 m/s², $\mu_s=0.5$, $\mu_k=0.4$, and apply a horizontal pull at a height chosen so that the block does not tip. With no vertical acceleration, $f_n=mg=19.62$ N. The static allowance is $0.5(19.62)=9.81$ N.

An 8 N pull permits $f_t=-8$ N: net horizontal force and acceleration are zero. A 12 N pull would require $f_t=-12$ N, outside the cone, so sticking is impossible. Once rightward sliding begins, kinetic friction is $-0.4(19.62)=-7.848$ N and acceleration is $(12-7.848)/2=2.076$ m/s². Static and kinetic coefficients are assumed fixture values, not measured material properties.

The diagram is force space, not a drawing of the block. Vertical position encodes normal force at 9 drawing units/N; horizontal position encodes tangent force at 12 units/N. Thus the drawn cone angle is intentionally rescaled, while its force coordinates still obey the inequality. The horizontal slice fixes 19.62 N. The solid vector ends at the feasible sticking force; the crossed dashed vector is the impossible force needed to hold against 12 N. Neither depicts the post-slip kinetic force or a trajectory.

How do planners use the cone?

Given a candidate contact mode, require each contact's forces to satisfy its cone, then balance the resulting net wrench. That is the next step in grasp wrench balance. In three dimensions, independently bounding both tangent components by $\mu f_n$ admits corner forces of magnitude $\sqrt2\mu f_n$, outside the true circular cone. An inscribed friction pyramid sacrifices some feasible directions for linear inequalities. A circular-cone constraint preserves that boundary but changes the optimization problem.

MIT's contact equilibrium formulation combines unilaterality, friction and force/moment balance. A compliant normal law instead permits deformation and ties force to it. Compliance can better represent soft pads but needs stiffness and damping calibration. A rigid Coulomb model is a useful first screening for dry contact; adhesive, lubricated or strongly deforming interfaces need a different law. Impact impulses also need a collision model, rather than applying these static equations across an instantaneous velocity jump.

These contact directions also determine which quantities a controller can regulate. Hybrid force/position control uses the surface tangent for motion and its normal for pressing force, with explicit treatment of contact establishment and loss.

Code

This standalone Python makes the mode decision for an initially stationary block. The pull is positive and horizontal, with tipping excluded. A second case shows why the static boundary must be distinguished from actual friction.

1def initial_response(pull: float, mass: float = 2.0) -> tuple[str, float, float]:2    if pull < 0 or mass <= 0:3        raise ValueError("Requires a nonnegative pull and positive mass")4    normal = mass * 9.815    if pull <= 0.5 * normal:6        return "stick", -pull, 0.07    friction = -0.4 * normal8    return "slide", friction, (pull + friction) / mass9
10
11for pull in (8.0, 12.0):12    mode, friction, acceleration = initial_response(pull)13    print(f"{pull:.0f} N: {mode}, friction {friction:.3f} N, a {acceleration:.3f} m/s^2")14assert initial_response(0.0) == ("stick", -0.0, 0.0)15assert initial_response(8.0)[1] != -0.5 * 2.0 * 9.81

Output is 8 N: stick, friction -8.000 N, a 0.000 m/s^2 and 12 N: slide, friction -7.848 N, a 2.076 m/s^2.

Watch Out For

The Quick Version

• A nonadhesive contact pushes normally and cannot transmit normal force across a gap.
• Sticking friction occupies a bounded set; sliding friction opposes relative slip.
• The example's static allowance is 9.81 N, so 8 N sticks and 12 N initiates sliding.
• Three-dimensional tangential force must satisfy a magnitude bound, not two independent maxima.
• Contact laws constrain forces; full dynamics and geometry determine the motion.

What to Read Next

• Robot Dynamics: Forces, Inertia and Torque
• Robot Grasp Wrench Balance and Force Closure
• Robot Impedance Control: Yielding Under Contact
• Legged Robot Balance and the Capture Point
• Robot Simulation and Physics