Friction Model

Coulomb friction models the constant force that opposes motion, independent of velocity. It is characterized by distinct static and kinetic coefficients.

• Static Friction (Stiction): The force that must be overcome to initiate motion from rest. It is typically higher than kinetic friction.
• Kinetic Friction: The constant opposing force present during sliding motion.
• Model: Often represented as F_c = μ * F_n * sign(v), where μ is the friction coefficient and F_n is the normal force.

Viscous friction models the force that increases linearly with the relative velocity between surfaces, analogous to moving through a fluid.

• Linear Damping: The force is proportional to velocity (F_v = b * v), where b is the viscous damping coefficient.
• Energy Dissipation: This component is responsible for dissipating kinetic energy, preventing infinite oscillation in simulated systems.
• Application: Critical for modeling motor damping, air resistance on moving parts, and the behavior of lubricated bearings.

The Stribeck effect captures the observed decrease in friction force as velocity increases from zero, bridging static and kinetic friction regimes.

• Negative Damping: At very low velocities, friction force decreases with increasing speed before the viscous term dominates.
• Realism: This non-linear transition is essential for accurately simulating the 'stick-slip' phenomenon and the startup behavior of motors and sliding contacts.
• Model: Often incorporated via an exponential decay term added to the Coulomb model.

This component specifically models the discontinuity at zero velocity, where the applied force must exceed a threshold to overcome static adhesion.

• Breakaway Torque: The maximum torque a motor can output before its rotor begins to turn against static friction.
• Hysteresis: The model must account for the fact that the breakaway force is not applied continuously; once motion starts, the resisting force drops to the kinetic friction level.
• Implementation: Often handled with a conditional state machine in the simulation, tracking whether the system is in a 'stuck' or 'sliding' state.

Advanced models incorporate dependencies beyond velocity, making friction a function of multiple system states for higher fidelity.

• Position-Dependent Friction: Accounts for variations due to joint angle, lead screw position, or uneven wear on a surface.
• Load-Dependent Friction: Models how normal force (F_n) changes with payload or configuration, directly scaling Coulomb friction.
• Temperature Dependence: Simulates how heating in a motor or gearbox can reduce lubricant viscosity, altering viscous damping coefficients during extended operation.

The friction model is not standalone; it is integrated into the broader simulation loop to affect system dynamics.

• Actuator Model Integration: The computed friction torque is subtracted from the motor's commanded torque to determine the net torque applied to the joint.
• Physics Engine Coupling: Provides the resistive forces for the rigid body dynamics solver during contact resolution between objects.
• Numerical Stability: The discontinuous nature of Coulomb friction requires careful numerical integration techniques (like implicit solvers) to prevent simulation instability.

The primary role of a friction model is to accurately simulate the non-linear resistive forces in joints and actuators. Without it, a policy trained in a frictionless simulation will fail on real hardware because:

• It overestimates the robot's ability to make precise, small movements (defeated by static friction or stiction).
• It cannot manage limit cycles—small, persistent oscillations—caused by the stick-slip phenomenon.
• It misjudges energy dissipation, leading to overshoot and instability. A calibrated model allows the policy to learn robust control strategies that account for these real-world dynamics.

Incorporating friction during training forces the reinforcement learning agent to develop generalized, noise-invariant strategies. This is often combined with Domain Randomization, where friction parameters (Coulomb, viscous coefficients) are varied within plausible bounds. The policy learns to:

• Exert slightly higher initial torques to overcome stiction.
• Apply damping to settle movements smoothly.
• Be less sensitive to exact actuator dynamics, which vary between units and degrade with wear. This results in a policy that is robust to the parameter uncertainty inherent in all physical systems.

Before simulation training begins, real robot data is used to calibrate the friction model. This process, called System Identification (SysID), involves:

• Commanding the physical joint through a range of motions and measuring its response with high-resolution encoders and torque sensors.
• Fitting the parameters of a chosen friction model (e.g., LuGre, Maxwell-slip) to the recorded data.
• Validating the model by comparing its predicted motion to new, unseen real-world motion sequences. A well-identified model is the foundation for a high-fidelity simulation.

An oversimplified friction model (or none at all) introduces a simulation bias. The policy exploits this bias, learning behaviors that are physically impossible. For example, it might learn to rely on infinitesimally small joint adjustments for balance—a strategy that fails instantly on hardware where stiction prevents such micro-movements. A high-fidelity model mitigates this by ensuring the action space and state transitions in simulation closely match the real world's response, leading to policies that transfer effectively.

Different mathematical models capture friction at varying levels of fidelity and computational cost:

• Coulomb + Viscous: A simple, static model combining constant friction force and velocity-dependent damping. Fast but misses stiction and pre-sliding displacement.
• LuGre Model: A dynamic model that captures the bristle deflection analogy, simulating pre-sliding displacement, stiction, and the Stribeck effect. More computationally expensive but highly accurate for precise control.
• Maxwell-slip Models: Use a set of parallel spring-slider elements to model hysteresis and non-local memory effects. The choice depends on the required precision and available compute for real-time simulation.

The friction model interacts directly with the robot's control stack in simulation:

• Feedforward Compensation: The learned policy or controller can output torques that explicitly counteract the predicted friction force.
• Impedance & Force Control: Fidelity is critical for simulating compliant interactions. A poor model makes a robot seem either too sticky or too slippery during contact tasks.
• Joint-Level PID Tuning: Simulated PID gains tuned without friction will be wildly inappropriate on hardware, often causing instability. The model allows for realistic controller tuning in silico.

An actuator model is a mathematical representation of a physical motor's dynamics, defining its response to input commands. It encompasses electrical characteristics, mechanical limitations, and non-linear effects. A comprehensive actuator model integrates a friction model as a core sub-component to accurately simulate the resistive forces opposing motion.

• Core Components: Motor constants, winding resistance, inductance, back-EMF, rotor inertia, and torque/speed saturation limits.
• Integration with Friction: The total torque output is often calculated as: Commanded Torque = Modeled Torque - Friction Torque(velocity, load). This allows simulation of stiction, where the motor must overcome static friction to initiate movement.

Torque control is a low-level actuation mode where a motor driver directly commands a desired output torque or current. This mode enables compliant, force-sensitive robot behavior but is highly sensitive to unmodeled dynamics like friction. An accurate friction model is essential for high-performance torque control, as the controller must compensate for friction losses to deliver the intended force at the joint.

• Direct Drive Actuators: Often use torque control for their backdrivability, making precise friction compensation critical.
• Friction Compensation: Advanced controllers use a feedforward term based on the friction model to cancel out predicted resistive torque before it affects tracking error.

Inverse dynamics is the computation of the joint torques or forces required to achieve a desired acceleration of a robotic system, given its kinematic structure and mass distribution. This calculation is fundamental for model-based control. A friction model is added to the inverse dynamics solution to determine the actual motor torque required, as part of the total load: τ_motor = τ_inertial + τ_gravity + τ_coriolis + τ_friction. Without this term, commanded torques will be insufficient, causing lag and steady-state error in trajectory tracking.

System identification (SysID) is the experimental process of building mathematical models of dynamic systems from measured input-output data. In robotics, SysID is used to calibrate simulation parameters, including those for friction models, to match real hardware behavior. This bridges the sim-to-real gap.

• Procedure: The robot executes specially designed excitation trajectories while recording joint positions and motor currents/torques.
• Parameter Estimation: Algorithms like least-squares regression fit the data to a structured model (e.g., a Dahl or LuGre friction model) to identify static, Coulomb, and viscous friction coefficients.

Proprioception is a robot's sense of its own body's position, orientation, and movement, derived from internal sensors. Accurate proprioception is degraded by unmodeled friction, which introduces discrepancies between commanded motion and actual motion. Simulating realistic proprioceptive feedback requires injecting the effects of a friction model into sensor readings.

• Affected Sensors: Joint encoders, torque sensors, and inertial measurement units (IMUs).
• Simulation Integration: The simulator uses the friction model to compute the actual joint velocity and torque, which are then passed through a sensor noise model to generate synthetic encoder and torque readings for the control policy.

A physics material defines the surface properties of a simulated object that affect physical interactions, such as coefficients of friction and restitution (bounciness). While a joint/actuator friction model governs internal resistance, physics materials define external contact friction between the robot and its environment.

• Parameters: Static friction coefficient, dynamic (kinetic) friction coefficient, and friction combine mode.
• Hierarchical Modeling: A full simulation requires both: 1) Internal friction models for each joint's drive train, and 2) Physics materials on the robot's feet, fingers, or wheels to simulate environmental interaction. The Coulomb friction concept is common to both.