Contact mechanics and friction

Every grasp is held together by contact forces. To reason about whether a grasp will hold, how much squeeze is needed, or why an object slips and twists, we need simple but useful models of what happens where a fingertip touches an object. This lesson develops those models: contact types, Coulomb friction and the friction cone, its linearized approximation, soft-finger contacts that resist twisting, and what happens when contacts slip. The tools here feed directly into the grasp analysis of lesson 4 and the force sizing of lesson 7.

Contact forces and wrenches

When a fingertip presses on an object, it exerts a distributed pressure over a small contact patch. For analysis we usually replace that distribution with a single force acting at a contact point, plus in some models a moment. The force can be split into a normal component \( f_n \), perpendicular to the surface and pointing into the object, and a tangential component \( f_t \) in the surface plane, which is friction.

Contact forces can only push, never pull, unless there is adhesion or suction. This unilateral constraint \( f_n \ge 0 \) is what makes grasping different from bolting two parts together. A finger can always lose contact by moving away, and the grasp must be designed so that every contact keeps a positive normal force under all expected loads.

A force applied at a point also creates a moment about the object's reference point. Combining force and moment gives a wrench, a six-dimensional vector in 3-D (three force components and three moment components) or a three-dimensional vector in the plane (two force components and one moment). If the contact force \( \mathbf{f} \) acts at position \( \mathbf{p} \) relative to the object's reference point, the moment is \( \mathbf{p} \times \mathbf{f} \), and the wrench is

Grasp analysis asks whether the set of wrenches the contacts can apply is enough to balance any external wrench, such as gravity or an acceleration load. Lesson 4 builds that analysis on the contact models below.

Contact models

Three contact models cover most grasp analysis.

A frictionless point contact can only apply a force along the surface normal. It is a useful worst case and describes contacts on very slippery surfaces. One wrench direction is available, scaled by \( f_n \ge 0 \).

A point contact with friction (also called a hard-finger contact) can apply a normal force plus a tangential friction force, as long as the friction force stays within the limit set by Coulomb's law. In 3-D this gives three force components; in the plane, two.

A soft-finger contact adds a limited moment about the surface normal, because a deformable fingertip has a finite contact patch that can resist twisting. This torsional friction is what lets two rubber fingertips hold a pen against rotation about the line joining the fingers. A hard point contact cannot do that.

Coulomb friction and the friction cone

Coulomb's model of dry friction says that while two surfaces stick, the tangential force can take any value up to a limit proportional to the normal force:

Here \( \mu \) is the coefficient of static friction. If the tangential force needed to hold the object would exceed \( \mu f_n \), the surfaces slip, and the friction force then acts opposite to the relative sliding velocity with magnitude \( \mu_k f_n \), where \( \mu_k \), the kinetic coefficient, is usually somewhat smaller than \( \mu \).

Geometrically, the set of contact forces that do not cause slip forms a cone around the surface normal. Its half-angle is

Any force inside or on the cone is achievable without slip; any force outside it is not. A larger friction coefficient gives a wider cone and a more forgiving grasp. Typical values vary widely: rubber or polyurethane pads on cardboard or painted metal might have \( \mu \) around 0.5 to 1.0 when clean and dry, steel on steel around 0.15 to 0.3 when lightly oiled, and smooth plastic on plastic around 0.2 to 0.4. Real values depend strongly on contamination, humidity, wear and pressure, so grasp design uses conservative values and safety factors.

Linearizing the cone

The friction cone is a curved surface, which makes many grasp computations nonlinear. A common simplification replaces the cone with a pyramid of \( m \) edges. Any force inside the pyramid is a non-negative combination of its edge vectors. With four edges the pyramid can either inscribe the cone (conservative, slightly smaller) or circumscribe it (optimistic). Eight or more edges give a close approximation. In the plane the cone is just two edge vectors, so the planar case needs no approximation at all.

Writing contact forces as non-negative combinations of edge vectors turns many grasp questions into linear programs or convex hull tests, which are fast and reliable. In 3-D with four edges per contact, each contact contributes four wrench vectors; a three-finger grasp contributes twelve. Lesson 4 uses exactly this representation.

# Inscribed linearized friction cone: edge vectors and the worst-case loss of friction
import numpy as np
mu = 0.5
for m in (4, 6, 8, 16):
    ang = 2 * np.pi * np.arange(m) / m
    edges = np.stack([mu * np.cos(ang), mu * np.sin(ang), np.ones(m)], axis=1)  # normal along z
    # inscribed pyramid: the weakest direction lies midway between edges
    mu_eff = mu * np.cos(np.pi / m)
    print(f"m={m:2d}: edges={edges.shape[0]}, worst-case effective mu = {mu_eff:.3f} ({100*mu_eff/mu:.1f} % of mu)")

m= 4: edges=4, worst-case effective mu = 0.354 (70.7 % of mu)
m= 6: edges=6, worst-case effective mu = 0.433 (86.6 % of mu)
m= 8: edges=8, worst-case effective mu = 0.462 (92.4 % of mu)
m=16: edges=16, worst-case effective mu = 0.490 (98.1 % of mu)

The output shows the price of linearization. With only four edges, the weakest direction of an inscribed pyramid allows about 71 % of the true friction; with eight edges, about 92 %. Conservative grasp analysis accepts that loss, because underestimating friction is safer than overestimating it.

Soft-finger contact and torsional friction

A deformable fingertip pressed against a surface forms a contact patch of radius \( a \). Friction distributed over the patch can resist a moment \( m_z \) about the normal. A common model, for a uniform pressure distribution, gives a maximum torsional moment

Other pressure distributions change the constant: Hertzian contact gives about \( \tfrac{3\pi}{16} \mu a f_n \approx 0.59 \mu a f_n \). A simple engineering approach treats the soft-finger contact as an elliptical limit surface combining tangential force and torsion: when the contact is also carrying a large tangential force, less torsional capacity remains, and vice versa.

Slip, sliding and stick-slip

When a contact slips, the friction force drops to the kinetic value and points against the sliding direction. In grasping, slip is usually a failure, but controlled slip is useful: a robot can let a held object slide down between the fingers to reposition it, or pivot an object about a fingertip. Lesson 9 calls these in-hand manipulation primitives.

Stick-slip occurs when static friction is noticeably larger than kinetic friction and the system has compliance. The contact sticks, elastic energy builds up, the contact suddenly slips, the force drops, and the cycle repeats as a jerky motion. It can cause chatter in insertion tasks and noise in sliding mechanisms. Lubrication, materials with similar static and kinetic coefficients, stiffer structures and velocity control help.

Slip can be detected before an object falls. Tactile sensors see the contact patch shear and the edges start to slide, vibration sensors pick up high-frequency signals, and wrist force-torque sensors see the tangential-to-normal ratio approaching \( \mu \). Grip force controllers respond by increasing normal force as soon as incipient slip appears, which lets the robot hold fragile objects with minimal squeeze.

Contact stiffness, area and pressure

Real contacts deform. Hertz contact theory relates normal force to contact patch size and penetration for curved elastic bodies; for a sphere on a flat, the contact radius grows with the cube root of force. Two practical consequences follow. First, soft pads spread force over a larger area, lowering peak pressure, which protects fragile objects. Second, contact stiffness affects control: a stiff contact between a rigid tool and a rigid part turns tiny position errors into large forces, which is why impedance control or passive compliance is essential for insertion.

Pressure limits are often the binding constraint in food and pharmaceutical handling. Strawberries, eggs, bread and blister packs bruise or crush above certain pressures. Designers spread force over a larger contact area with conformal pads, add more contacts, or switch to suction or soft grippers.

Uncertainty in friction

Because friction is uncertain, robust grasp design treats \( \mu \) as a range. A designer might measure friction on samples of the actual product with the actual pad material, using a simple tilt test (the angle at which an object starts to slide on an inclined pad gives \( \mu = \tan\theta \)) or a pull test with a force gauge. They then design to the lower end of the measured range and add a safety factor. Contaminants in the real environment, such as dust, oil, condensation or release agents on moulded parts, should be included in the tests.