Coriolis and Centrifugal Terms

The torque terms in a robot's equations of motion that depend on the square or product of joint velocities.

Manipulator dynamics is commonly written as M(q)q̈ + C(q,q̇)q̇ + g(q) = τ: q is the joint angles, q̇ and q̈ are joint velocity and acceleration, M is the mass matrix, g is the gravity term, and τ is joint torque. The term C(q,q̇)q̇ is the Coriolis and centrifugal term. Modern Robotics calls the part that depends only on the square of a single joint velocity, q̇ᵢ², the centrifugal term, and the part depending on the product of two different joint velocities, q̇ᵢq̇ⱼ, the Coriolis term. It arises because joint coordinates aren't an inertial frame: even when every joint spins at constant velocity (q̈ = 0), the links are still moving in circles, which requires torque to sustain. At low speed this term is small and often ignored, but at high speed, failing to compensate for it produces visible tracking error. Inverse dynamics and computed-torque control both need to compute it, and the skew-symmetry property of Ṁ − 2C is a standard tool for proving controller stability.

ExampleIn a planar two-link arm with the second joint bent to 90°, if both joints rotate forward at constant angular velocity simultaneously, the end-effector mass is pulled closer to joint 1, and joint 1 actually needs to output a negative torque to hold its speed steady — this is the Coriolis term at work.

Also called

Coriolis Matrix, C(q, q̇), Nonlinear Velocity Terms

Related

Euler-Lagrange Equations · Mass Matrix · Rigid-Body Dynamics · Inverse Dynamics · Computed Torque Control · Recursive Newton-Euler Algorithm

Sources

Lynch & Park, Modern Robotics（Ch. 8 Dynamics of Open Chains）
Wikipedia: Coriolis force