Friction Cone

• Also called: Coulomb friction cone, friction pyramid, linearized friction cone
• Category: Planning & manipulation › Manipulation
• Tags: contact-mechanics, friction, grasping, legged-locomotion, optimization

The set of contact forces a surface can transmit without slipping under Coulomb friction, shaped as a cone around the surface normal whose half-angle is set by the friction coefficient.

Definition

Friction cone is the geometric description of which forces a contact can transmit without sliding. Under the Coulomb friction model, a contact can resist a tangential force up to the friction coefficient times the normal force. If the normal force is f_n and the tangential force is f_t, the contact sticks as long as |f_t| ≤ μ f_n, where μ is the coefficient of static friction. Plotted in three dimensions, the set of allowable force vectors forms a cone whose axis is the surface normal and whose half-angle is arctan(μ).

Any force inside the cone can be applied without slipping; a force on the boundary is on the verge of slip; a force outside the cone cannot be sustained, and the contact slides. A contact can also only push, not pull, so the normal component must be non-negative.

The friction cone appears throughout manipulation and locomotion. In grasp analysis, the friction cones at the fingertips determine which forces and torques on the object a grasp can resist, which is the basis of force-closure tests. In legged locomotion, every foot's ground reaction force must stay inside its friction cone, or the foot slips, and planners and controllers enforce this as a constraint.

Because the cone is a nonlinear (second-order cone) constraint, many optimization-based controllers replace it with a friction pyramid, a polyhedral approximation with four or more faces, which turns the constraint into linear inequalities suitable for quadratic programming. A conservative friction coefficient is usually chosen, since real friction varies with surface, material and contamination.

Why it matters

Whenever a robot grasps, pushes or walks, it relies on friction, and the friction cone is the constraint that tells planners which forces are safe to apply. Violating it causes dropped objects and slipping feet, so it appears in nearly every model-based grasp planner and legged locomotion controller.

Key applications

• Force-closure and grasp quality analysis
• Ground reaction force constraints in legged locomotion controllers
• Contact-rich manipulation planning such as pushing and pivoting
• Multi-contact motion planning for humanoids
• Contact modeling in physics simulators

In practice

Control engineers working on a quadruped or humanoid add friction cone constraints, usually linearized as pyramids, to the quadratic program or model predictive controller that computes foot forces, using a friction coefficient lower than the value expected for the floor to leave a margin. Grasp planners do the same at each fingertip contact when checking whether a grasp can hold an object against gravity and expected disturbances. When a robot slips in testing, teams first check whether the commanded forces were close to the cone boundary and whether the assumed friction coefficient was realistic for that surface.

Common misconceptions

• The friction coefficient is a fixed property of one material — it depends on the pair of surfaces in contact and on conditions such as dust, moisture and wear
• A friction pyramid is exact — it approximates the cone; an inscribed pyramid is conservative and slightly restricts the usable forces
• Staying inside each friction cone guarantees a stable grasp — every contact must stick, but the contacts together must also be able to balance the load on the object

Who should know this

• Legged locomotion control engineers
• Grasp and manipulation planning researchers
• Physics simulation developers
• Mechanical engineers designing grippers and robot feet

Related technologies

• Coulomb Friction
• Force Closure
• Quadratic Programming
• Whole-Body Control
• Contact Dynamics
• Grasp Wrench Space

Related terms

• Grasp Planning
• Whole-Body Control
• Gripper
• Centroidal Dynamics

Related projects

• Drake: MIT robotics toolbox for model-based dynamics, optimization, and control system design
• GTSAM: Factor-graph optimization library powering many modern SLAM backends
• PyPose: PyTorch library for differentiable robotics with deep learning and Lie group optimization
• NeuPAN: Real-time end-to-end planner that maps raw lidar points directly to robot actions
• OpEn (Optimization Engine): Embedded nonconvex optimization solver in Rust for real-time MPC on robots
• ProxSuite: High-performance QP/LP solver for robotics control and optimization

Videos

• Coulomb Friction, Cone of Friction, Falling Broom & Spinning Top | ESM 5314 Lecture 25: Dr. Shane Ross
• Modern Robotics, Chapter 12.2.1: Friction: Northwestern Robotics