11.7.1. Friction Models

• Static Friction (Coulomb): Constant opposing force at low velocity

Static friction, often referred to as Coulomb friction, is defined as the force that opposes the start of motion between two surfaces that are in contact and at rest relative to each other. This force remains constant until a certain threshold of applied force is surpassed, at which point the object will begin to move. At low velocities, static friction is able to counteract small forces without movement occurring. Once the applied force exceeds the maximum static friction, motion will begin.

Imagine trying to push a heavy box on the floor. Initially, there's a constant force resisting your push—the static friction—holding the box in place. Only when you push hard enough to overcome this static friction does the box slide across the floor.

• Viscous Friction: Proportional to velocity

Viscous friction is the type of friction that opposes the motion of an object in a fluid (which could be a liquid or gas) and increases with the speed of that motion. Mathematically, it is represented as a force that is directly proportional to the velocity of the moving object. Essentially, the faster you move through a medium (like water or air), the greater the resistive force you encounter due to this type of friction.

When you try to swim through water, you feel a resistance that gets stronger as you swim faster. This is viscous friction; the more quickly you try to move through the water, the more resistance you face, which can make swimming harder.

• Stribeck Effect: Drop in friction near zero velocity

The Stribeck effect describes a phenomenon where friction decreases as the velocity of an object approaches zero. Initially, as speed reduces, static friction is dominant; however, as the velocity nears zero, the effective friction reduces, transitioning into viscous behavior. This leads to a drop in frictional resistance, which can be observed in lubricated systems where a film of lubricant forms at lower speeds, allowing for smoother operation.

Think of riding a bicycle. When you are moving at speed, you can feel some resistance against the wheels due to friction. However, when you slow almost to a stop, you may notice that it becomes easier to keep the bike rolling. This is because the frictional force decreases when very little motion is occurring, thus resembling the Stribeck effect.

Total torque due to friction: τ = τ_f · sign(q˙) + b · q˙

The total torque exerted by friction is a combination of two components: the first component τ_f, which is influenced by the direction of motion (given by sign(q˙), indicating the direction), and the second component which depends on velocity (b · q˙, with b being a constant related to viscous friction). This equation allows for calculating the net torque experienced by a robot or mechanical system due to frictional forces, considering both static and viscous contributions.

When you pull a drawer, the amount of torque you feel varies with both the speed at which you pull and the resistance of the drawer due to friction. Initially, the static friction holds it in place until you pull hard enough, then moving it encounters both the change in direction and resistance, described by this equation.

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